Conic Sections Class 10 Notes CBSE Maths Chapter 10 (Free PDF Download)

Section–A (1 Mark Questions)

1. The equation of circle, which passes through origin and has centre at (–1, 4) is ________.

Ans.  We know that,

Equation of circle with centre (h, k) and passes through the origin is

$(x-h{)}^2+(y-k{)}^2=h^2+k^2$

Thus, the equation of required circle with centre (-1, 4) is-

$(x+1{)}^2+(y-4{)}^2=(-1{)}^2+4^2$

$\Rightarrow x^2+2x+1+y^2-8y+16=1+16$

$\Rightarrow x^2+y^2+2x-8y=0$

2. For parabola y2=-16x, find the equation of directrix.

Ans. Given: Equation of parabola: y2=-16x 

Since the given parabola is of the form y2=-4ax 

Thus, on comparison we get,

16=4a

$\Rightarrow$ a=4

Therefore,

Equation of directrix: x = a i.e., x = 4.

3. For the ellipse $\frac{x^2}{81}+\frac{y^2}{9}=1$, find the coordinates of end points of major axis.

Ans. Equation of ellipse: 

$\frac{x^2}{81}+\frac{y^2}{9}=1$

Since denominator of x2  > denominator of y2

$\therefore$ major axis: x-axis and minor axis: y-axis.

$\therefore$ On comparing the given equation with $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$  

We get, a = 9 and b = 3

Thus, Coordinates of end points of major axis are $(\pm a,0)$ i.e., $(\pm 9,0)$ 

4. For the hyperbola $\frac{y^2}{16}-\frac{x^2}{4}=1$, find the length o transverse axis.

Ans. Equation of hyperbola: $\frac{y^2}{16}-\frac{x^2}{4}=1$ 

Since the given hyperbola is of the form \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 

Thus, on comparison we get, a = 4 and b = 2

Therefore, 

Length of transverse axis $=2a=2\times =8$.

5. Find the equation of parabola with vertex (0, 0) and focus (0, -7)

Ans. Since the vertex is at (0, 0) and focus is at (0, –7), i.e., on y -axis

Thus,  y - axis is the axis of parabola.

Hence, the equation of the parabola is of the form $x^2=-4ay$ with a = 7.

Therefore, equation of parabola is given by

$x^2=-4ay=-4(7)y$

$\Rightarrow x^2=-28y$

Section–B (2 Marks Questions)

6. Find the equation of the circle which touches the y-axis and centre (2, 4).

Ans. Since, the circle is touching y-axis. 

The length CP is equal to the radius of the circle. 

Thus, Radius = CP = 2.

Hence, the equation of the required circle is

$(x-2{)}^2+(y-4{)}^2=2^2$

$\Rightarrow x^2-4x+4+y^2-8y+16=4$

$\Rightarrow x^2+y^2-4x-8y+16=0$

7. Find the length of latus rectum of the hyperbola $\frac{y^2}{12}-\frac{x^2}{4}=1$.

Ans. Equation of hyperbola: $\frac{y^2}{12}-\frac{x^2}{4}=1$

Since, the given hyperbola is of the form

$$\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$$

Thus, on comparison we get,

$$b^2=4,a^2=12\Rightarrow a=\sqrt{12}$$

Therefore, Length of latus rectum

$$=\frac{2b^2}{a}=\frac{2\times 4}{\sqrt{12}}=\frac{4}{\sqrt{3}}$$

8. Find the coordinates of the foci a of the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$.

Ans. Equation of hyperbola: $\frac{x^2}{16}-\frac{y^2}{9}=1$

On comparing with standard equation of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$,

We obtain $a=4$ and $b=3$.

Now, we know that,

$$\begin{array}{rl}& b^2=c^2-a^2\\ & \Rightarrow c^2=a^2+b^2\\ & \Rightarrow c^2=4^2+3^2=25\\ & \Rightarrow c=5\end{array}$$

Thus,Coordinates of foci are $(\pm c,0)$ i.e., $(\pm 5,0)$

9.Find the equation of the circle which touches the lines x = 0, y = 0 and x = -6.

Ans. Since, the circle is touching the lines $x=0,y=0$ and $x=-6$.

Thus, two such circles are possible

$C_1:(-3,3)$ and $r_1=3$

$C_2:(-3,-3)$ and $r_2=3$

Thus, the equation of circles is-

$$\begin{array}{rl}& (x+3{)}^2+(y\pm 3{)}^2=(3{)}^2\\ & \Rightarrow x^2+y^2+6x\pm 6y+9=0\end{array}$$

10.Find the equation of a circle with centre (1, 3) and passes through the point (-1, -3).

Ans. Centre of the circle $(h,k)=(1,3)$.

Since the circle passes through point $(-1,-3)$.

The radius $(r)$ of the circle is the distance between the points $(1,3)$ and $(-1,-3)$.

$\Rightarrow r=\sqrt{(1-(-1){)}^2+(3-(-3){)}^2}$

$=\sqrt{(2{)}^2+(6{)}^2}=\sqrt{4+36}=\sqrt{40}=2\sqrt{10}$

Thus, the equation of the circle is given by-

$$\begin{array}{rl}& (x-1{)}^2+(y-3{)}^2=(2\sqrt{10}{)}^2\\ & \Rightarrow x^2-2x+1+y^2-6y+9=40\\ & \Rightarrow x^2+y^2-2x-6y-30=0\end{array}$$

11. Find the equation of the circle, when the coordinates of end point of diameter is given by (1, 1) and (3, 3).

Ans. We know that equation of the circle with diametric points $(x_1,y_1)$ and $(x_2,y_2)$ is given as $(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0$

Thus, when the coordinates of end points of diameter are $(1,1)$ and $(3,3)$.

Equation of circle:

$$\begin{array}{rl}& (x-1)(x-3)+(y-1)(y-3)=0\\ & \Rightarrow x^2-4x+3+y^2-4y+3=0\\ & \Rightarrow x^2+y^2-4x-4y+6=0\end{array}$$

12. Find the equation of the ellipse having ends of major axis $(\pm 5,0)$ and ends of minor axis $(0,\pm 3)$.

Ans. Given: ends of major axis $(\pm 5,0)$ and ends of minor axis $(0,\pm 3)$.

Since, the major axis is along $x$ - axis, Therefore, equation of the ellipse will be the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, where $a$ is the semimajor axis.

Accordingly, $a=5$ and $b=3$.

Thus, required equation of ellipse is $\frac{x^2}{5^2}+\frac{y^2}{3^2}=1$ or $\frac{x^2}{25}+\frac{y^2}{9}=1$.

13. Find the coordinates of the focus and equation of the directrix for the parabola given by $y^2=-2x$.

Ans. Given: Equation of parabola: $y^2=-2x$ Since, the given parabola is of the form $y^2=-4ax$,

Thus, on comparison we get.

$$\begin{array}{rl}& 2=4a\\ & \Rightarrow a=\frac{1}{2}\end{array}$$

Therefore,Coordinates of focus:

$$(-a,0)=(-\frac{1}{2},0)$$

Equation of directrix: $x=a$ i.e., $x=\frac{1}{2}$

PDF Summary - Class 11 Maths Conic Sections (Chapter 10)

1. Conic Sections:

The locus of point which moves in a plane so that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed straight line

• The fixed straight line is called the Directrix

• The fixed point is called the Focus. 

• The constant ratio is called the Eccentricity denoted by e. 

• A point of intersection of a conic with its axis is called a Vertex.

• The line passing through the focus & perpendicular to the Directrix is called the Axis. 

2. Section of Right Circular Cone by Different Planes:

Figure shows right circular cone.

(i) Figure shows the section of a right circular cone by a plane passing through its vertex is a pair of straight lines passing through the vertex.

(ii)  Figure shows section of a right circular cone by a plane parallel to its base is a circle. 

(iii) Figure shows section of a right circular cone by a plane parallel to a generator of the cone is a parabola.

(iv) Below figure shows section of a right circular cone by a plane neither parallel to any generator of the cone nor perpendicular or parallel to the axis of the cone is an ellipse or hyperbola.

3D view:

3. General Equation of a Conic: Focal Directrix Property.

If focus of conic is $$\left(p,q\right)$$ and equation of directrix is $$lx+my+n=0$$,then the general equation of conic is:

$(l^2=m^2)[(x-p{)}^2+(y-q{)}^2)]=e^2(lx+my+n{)}^2\equiv a{x}^2+2hxy+b{y}^2+2gx+2fy=c=0$

4. Distinguishing Various Conics:

The nature of conic section depends upon value of eccentricity as well as the position of the focus and the directrix. So, there are two different cases:

Case 1: When the Focus Lies on the Directrix.

In this case, $$\mathrm{\Delta }\equiv abc+2fgh--a{f}^2--b{g}^2--c{h}^2=0$$

The general equation of a conic represents a pair of straight lines if:

$$e>1\equiv h^2>ab$$, Real and distinct lines intersecting at focus

$$e\text{ = }1\equiv h^2⩾ab$$, Coincident lines

$$e\text{ < }1\equiv h^2\text{ < }ab$$, Imaginary lines

Case 2: When the Focus Does not Lies on the Directrix.

A parabola 

An ellipse

A hyperbola

A rectangular hyperbola 

Parabola

Definition and Terminology

The locus of a point, whose distance from a fixed point called focus is equal to perpendicular distance from a fixed straight line called directrix is called Parabola.

Four standard forms of parabola are:

$$y^2=4ax;y^2=-4ax;x^2=4ay;x^2=-4ay$$

For parabola, $$y^2=4ax$$:

(i) Vertex is: $$(0,0)$$

(ii) Focus is: $$(a,0)$$

(iii) Axis is: $$y=0$$

(iv) Directrix is: $$x+a=0$$

• The distance of a point on the parabola from the focus is called focal distance.

• A chord of the parabola, which passes through the focus is called focal chord.

• A chord of the parabola perpendicular to the axis of the symmetry is called double ordinate.

• A double ordinate passing through the focus or a focal chord perpendicular to the axis of parabola is called the Latus Rectum.

For, $$y^2=4ax$$:

$$\Rightarrow$$Length of Latus rectum $$=4a$$

$$\Rightarrow$$Ends of Latus rectum are $$(a,2a)\mathrm{\&}(a,-2a)$$

Note:

• Perpendicular distance from focus to  directrix = half the latus rectum.

• Vertex is middle point of the focus & the point of intersection of directrix & axis.

• Two parabolas are said to be equal if they have the same latus rectum.

Position of a Point Relative to Parabola:

Point $$(x_1,y_1)$$lies inside, on or outside the parabola $$y^2=4ax$$depends upon the value of $${y_1}^2-4a{x}_1$$, whether it is positive, negative or zero.

Line and Parabola:

The line $$y=mx+c$$ meets parabola $$y^2=4ax$$ at:

• Two real points if $$a>mc$$

• Two coincident points if $$a=mc$$

• Two non real points if $$a<mc$$

Condition of Tangency is $$c={\displaystyle \frac{a}{m}}$$

Length of chord that line $$y=mx+c$$intercepts on parabola $$y^2=4ax$$ is:

$$\left({\displaystyle \frac{4}{m^2}}\right)\sqrt{a\left(1+m^2\right)\left(a-mc\right)}$$

Note: Length of focal chord making an angle $$\alpha$$ with the x-axis is $$4a\cos e{c}^2\alpha$$

Parametric Representation:

$$\left(a{t}^2,2at\right)$$ represents the co-ordinates of a point on the parabola is $$y^2=4ax$$ i.e. the equations $$x=a{t}^2,y=2at$$ at together represents the parabola with t being the parameter. 

The equation of a chord joining $$t_1\mathrm{\&}{t}_2$$is $$2x--\left(t_1+t_2\right)y+2a{t}_1{t}_2=0.$$

Note: If $$t_1\mathrm{\&}{t}_2$$are ends of the focal chord of the parabola$$y^2=4ax$$, then $$t_1{t}_2=-1$$

Hence, coordinates of extremities of parabola are $$\left(a{t}^2,2at\right)or\left({\displaystyle \frac{a}{t^2}},{\displaystyle \frac{-2a}{t}}\right)$$

Tangent to the Parabola $$y^2=4ax$$

$$(i)y{y}_1=2a\left(x+x_1\right)$$ at point $$(x_1,y_1)$$

$$(ii)y=mx+{\displaystyle \frac{a}{m}}\left(m\ne 0\right)$$ at $$\left({\displaystyle \frac{a}{m^2}},{\displaystyle \frac{2a}{m}}\right)$$

$$(iii)ty=x+a{t}^2$$at point $$\left(a{t}^2,2at\right)$$

Note: Point of intersection of the tangents at the point $$t_1\mathrm{\&}{t}_{2}is\left[a{t}_1{t}_2,a\left(t_1+t_2\right)\right]$$

Normal to the Parabola $$y^2=4ax$$

$$(i)y-y_1={\displaystyle \frac{-y_1}{2a}}\left(x-x_1\right)at(x_1,y_1)$$

$$(ii)y=mx-2am-a{m}^{3}at(a{m}^2,-2am)$$

$$(iii)y+tx=2at+a{t}^{3}at(a{t}^2,-2at)$$

Note: 

 (i) Point of intersection of normals at $t_1\mathrm{\&}{t}_2$, are, $\left[a\left(t_1^2+t_2^2+t_1{t}_2+2\right),-a{t}_1{t}_2\left(t_1+t_2\right)\right]$

(ii) If the normals to the parabola $y^2=4ax$ at the point $t_1$ meets the parabola again at the point $t_2$ then $t_2=-\left(t_1+{\displaystyle \frac{2}{t_1}}\right)$

(iii) If the normals to the parabola $y^2=4ax$ at the points $t_1\mathrm{\&}{t}_2$ intersect again on the parabola at the point ${\text{t}}_3$  then ${\text{t}}_1{\text{t}}_2=2,{\text{t}}_3=-\left({\text{t}}_1+{\text{t}}_2\right)$ and the line joining $t_1\mathrm{\&}{t}_2$ passes through a fixed point $(-2a,0)$

Pair of Tangents

The equation to the pair of tangents which can be drawn from any point $\left(x_1,y_1\right)$ to the parabola $y^2=4ax$ is given by: $S{\mathbf{S}}_1=T^2$ where:

$S\equiv y^2-4ax,S_1=y_1^2-4a{x}_1,T\equiv y{y}_1-2a\left(x+x_1\right)$

Director Circle

Locus of the point of intersection of the perpendicular tangents to a curve is called the director circle. For parabola $y^2=4ax$ it's equation is $\mathbf{x}+\mathbf{a}=0$ which is parabola's own directrix.

Chord of Contact

Equation to the chord of contact of tangents drawn from a point $P\left(x_1,y_1\right)$ is $y{y}_1=2a\left(x+x_1\right);$ (i.e., $T=0)$

Note: The area of the triangle formed by the tangents from the point $\left(x_1,y_1\right)\mathrm{\&}$the chord of contact is. ${\left(y_1^2-4a{x}_1\right)}^{3/2}÷2a$

Chord With a Given Middle Point

Equation of the chord of the parabola ${\text{y}}^2=4ax$ whose middle point is $\left(x_1,y_1\right)$ is:

$\mathbf{y}-{\mathbf{y}}_1={\displaystyle \frac{2\text{a}}{{\text{y}}_1}}\left(\mathbf{x}-{\mathbf{x}}_1\right)\equiv \text{T}={\text{S}}_1$

Ellipse

Standard Equation and Definitions:

Standard equation of an ellipse referred to its principal axes along the co-ordinate axes is

${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$ , where $a>b,b^2=a^2\left(1-e^2\right)$

Eccentricity: $e=\sqrt{1-{\displaystyle \frac{b^2}{a^2}}},(0<e<1)$

Foci: $S=(ae,0)\mathrm{\&}{S}^{\prime }=(-ae,0)$

Equations of Directrices: $x={\displaystyle \frac{\text{a}}{\text{e}}}\mathrm{\&}x=-{\displaystyle \frac{\text{a}}{\text{e}}}$

Major Axis: The line segment $A^{\prime }A$ in which the foci $S^{\prime }\mathrm{\&}S$ lie is of length $2a$ &  is called the major axis $(\text{a}>\text{b})$ of the ellipse. Point of intersection of major axis with directrix is called the foot of the directrix $(Z)$.

Minor Axis: The y-axis intersects the ellipse in the points ${\text{B}}^{\prime }=(0,-\text{b})$ and $\text{B}=(0,\text{b})$. The line segment ${\text{B}}^{\prime }\text{B}$ of length $2\text{b}(\text{b}<\text{a})$ is called the minor axis of the ellipse.

Principal Axis: The major and minor axis together are called principal axis of the ellipse.

Vertices: $A^{\prime }\equiv (-a,0)\mathrm{\&}A\equiv (a,0)$.

Focal Chord : A chord which passes through a focus is called a focal chord.

Double Ordinate: A chord perpendicular to the major axis is called a double ordinate.

Latus Rectum: The focal chord perpendicular to the major axis is called the  latus rectum.

Length of latus rectum (LL’) :

${\displaystyle \frac{2{\text{b}}^2}{\text{a}}}={\displaystyle \frac{{(\text{ minor axis })}^2}{\text{ majer axis }}}=2\text{a}\left(1-{\text{e}}^2\right)$

$=2e$ (distance from focus to the corresponding directrix).

Centre: The point which bisects every chord of the conic drawn through it is called the centre of the conic. $C=(0,0)$

The origin is the centre of the ellipse ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$

Note:

(i) If the equation of the ellipse is given as ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$ and nothing is mentioned, then the rule is to assume that $a>b$

(ii) If $b>a$ is given then y-axis will become major axis and $x$ axis will become the minor axis and all other points and lines will change accordingly.

Auxiliary Circle/Eccentric Angle:

A circle described on major axis as diameter is called the auxiliary circle. Let $Q$ be a point on the auxiliary circle $x^2+y^2=a^2$ such that QP  produced is perpendicular to the $x$-axis, then $P$& $Q$ are called as the Corresponding Points on the ellipse and the auxiliary circle respectively. $\theta$ is called the Eccentric angle of the point $\text{P}$ on the ellipse $(-\pi <\theta ⩽\pi )$.

Note:

• ${\displaystyle \frac{l(PN)}{l(QN)}}={\displaystyle \frac{b}{a}}={\displaystyle \frac{\text{ Semi major axis }}{\text{ Semi major axis }}}$

• If from each point of a circle perpendiculars are drawn upon a fixed diameter then the locus of the points dividing these perpendiculars in a given ratio is an ellipse of which the given circle is the auxiliary circle.

Parametric Representation:

The equations $x=a\cos \theta \mathrm{\&}y=b\sin \theta$ , together represent the ellipse ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$

Where $\theta$ is a parameter. Note that if $P(\theta )=$ (a cos$\theta ,b\sin \theta )$ is on the ellipse then$Q(\theta )=(a\cos \theta ,a\sin \theta )$ is on the auxiliary circle.The equation to the chord of the ellipse joining two. points with eccentric angles $\alpha$ and $\beta$ is given by

${\displaystyle \frac{x}{a}}\cos {\displaystyle \frac{a+\beta }{2}}+{\displaystyle \frac{y}{b}}\sin {\displaystyle \frac{a+\beta }{2}}=\cos {\displaystyle \frac{a-\beta }{2}}$

Position of a Point With Respect to Ellipse:

The point $p\left(x_1,y_1\right)$ lies outside, inside or on the ellipse, according as:

${\displaystyle \frac{x_1^2}{a^2}}+{\displaystyle \frac{y_1^2}{b^2}}-1>0$ (outside)

${\displaystyle \frac{x_1^2}{a^2}}+{\displaystyle \frac{y_1^2}{b^2}}-1<0$ (inside)

${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1=0$ (on)

Line and an Ellipse:

The line $y=mx+c$ meets the ellipse ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$ in two points real, coincident or imaginary according as ${\text{c}}^2\text{is}<,=\mathrm{\&}>{\text{a}}^2{\text{m}}^2+{\text{b}}^2$

Hence $y=mx+c$ is tangent to the ellipse ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$, if $c^2=a^2{m}^2+b^2$.

Tangents:

(a) Slope form $y=mx\pm \sqrt{a^2{m}^2+b^2}$ is tangent to the ellipse for all values of $\text{m}$

(b) Point form ${\displaystyle \frac{x{x}_1}{a^2}}+{\displaystyle \frac{y{y}_1}{b^2}}=1$ is tangent to the ellipse at $\left(x_1,y_1\right)$

(c) Parametric for ${\displaystyle \frac{xcos\theta }{a}}+{\displaystyle \frac{y\sin \theta }{b}}=1$ is tangent to the ellipse at the point $(acos\theta ,\text{b}\sin \theta )$.

Note:

(i) There are two tangents to the ellipse having the same m, i.e. there are two tangents parallel to any given direction. These tangents touch the ellipse at extremities of a diameter.

(ii) Point of intersection of the tangents at the point $\alpha \mathrm{\&}\beta$ is:

$\left(a{\displaystyle \frac{\cos {\displaystyle \frac{\alpha +\beta }{2}}}{\cos {\displaystyle \frac{\alpha -\beta }{2}}}},b{\displaystyle \frac{\sin {\displaystyle \frac{\alpha +\beta }{2}}}{\cos {\displaystyle \frac{\alpha -\beta }{2}}}}\right)$

(iii) The eccentric angles of point of contact of two parallel tangents differ by$\pi$.

Normal:

(i) Equation of the normal at$\left(x_1,y_1\right)$ is ${\displaystyle \frac{a^{2}x}{x_1}}-{\displaystyle \frac{b^{2}y}{y_1}}=a^2-b^2=a^2{e}^2$

(ii) Equation of the normal at the point (acos$\theta$ , bsin$\theta$ ) is $\mathrm{ax}\sec \theta -\text{ by }\mathrm{cosec}\theta =\left(x^2-b^2\right)$

(iii) Equation of a normal in terms of its slope m is $y=mx+{\displaystyle \frac{\left(a^2-b^2\right)m}{\sqrt{a^2+b^2{m}^2}}}$

Director Circle:

Locus of the point of intersection of the tangents which meet at right angles is called the Director Circle. The equation to this locus is ${\text{x}}^2+{\text{y}}^2={\text{a}}^2+{\text{b}}^2$ i.e. a circle whose centre is the centre of the ellipse & whose radius is the length of the line joining the ends of the major & minor axis.

Note:

Pair of tangents, Chord of contact, Pole & Polar, Chord with a given Middle point are to be interpreted as they, are in Parabola/Circle.

Diameter (Not in Syllabus)

The locus of the middle points of a system of parallel chords with slope 'm' of an ellipse is a straight line passing through the centre of the ellipse, called its diameter and has the equation $y=-{\displaystyle \frac{{\text{b}}^2}{a^{2}m}}x$

Note:

All diameters of ellipse passes through its centre

Important Highlights:

Referring to the ellipse ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$

(a) If $\text{P}$ be any point on the ellipse with S & S’ as to foci then $SP+S{P}^{\prime }=2a$.

(b) The tangent & normal at a point $\text{P}$ on the ellipse bisect the external and internal angles between the focal distances of P. This refers to the well known reflection property of the ellipse which states that rays from one focus are reflected through other focus & vice-versa. Hence we can deduce that the straight lines joining each focus to the foot of the perpendicular from the other focus upon the tangent at any point $P$ meet on the normal PC and bisects it where $\text{G}$ is the point where normal at $P$ meets the major axis.

(c) The product of the length's of the perpendicular segments from the foci on any tangent to the ellipse is ${\text{b}}^2$ and the feet of these perpendiculars lie on its auxiliary circle and the tangents at these feet to the auxiliary circle meet on the ordinate of $\text{P}$ and that the locus of their point of intersection is a similar ellipse as that of the original one.

(d) The portion of the tangent to an ellipse between the point of contact & the directrix subtends a right angle at the corresponding focus.

(c) If the normal at any point $\text{P}$ on the ellipse with, centre $C$ meet the major and minor axes in $G$ & $g$ respectively &  if CF be perpendicular upon this normal then:

(i) PF. $\text{PG}={\text{b}}^2$

(ii) $\text{PF}.\text{Pg}={\text{a}}^2$

(iii) $\text{PG}\text{.Pg}=\text{SP}\text{.S'P}$

(iv) $\text{CG}.C\text{T}=\text{C}{\text{S}}^2$

(v) locus of the mid point of $\text{Gg}$ is another ellipse having the same eccentricity as that of the original ellipse.

(Where $S$ and $S^{\prime }$ are the foci of the ellipse and $\text{T}$ is the point where tangent at $\text{P}$ meet the major axis)

• The circle on any focal distance as diameter touches the auxiliary circle. Perpendiculars from the centre upon all chords which join the ends of any perpendicular diameters of the ellipse are of constant length.

• If the tangent at the point P of a standard ellipse meets the axis in T and t and CY is the perpendicular on it from the centre then :

     (i) T t.P Y = a2 – b2 and

           (ii) least value of T t is a+b

Hyperbola 

The Hyperbola is a conic whose eccentricity is greater than unity $(e>1)$

Standard Equation and Definitions:

Standard equation of the hyperbola is ${\displaystyle \frac{x^2}{a^2}}-{\displaystyle \frac{y^2}{b^2}}=1$, where $b^2=a^2\left(e^2-1\right)$

Eccentricity (e):  ${\text{e}}^2=1+{\displaystyle \frac{{\text{b}}^2}{{\text{a}}^2}}=1+{\left({\displaystyle \frac{\text{C}\cdot \text{A}}{\text{T}.\text{A}}}\right)}^2$

(C.A $\to$ Conjugate Axis)

(T.A. $\to$Transverse Axis)

Foci:  $S=(ae,0)\mathrm{\&}{S}^{\prime }=(-ae,0)$

Equations of Directrix: $\text{x}={\displaystyle \frac{\text{a}}{\text{e}}}\mathrm{\&}x=-{\displaystyle \frac{\text{a}}{\text{e}}}$

Transverse Axis: The line segment $A^{\prime }A$ of length 2a in which the foci $S^{\prime }\mathrm{\&}S$ both lie is called the transverse axis of the hyperbola.

Conjugate Axis: The line segment B'B between the two points $B^{\prime }=(0,-\text{b})\mathrm{\&}B=(0,\text{b})$ is called as the conjugate axis of the hyperbola.

Principal Axes: The transverse & conjugate axis together are called Principal Axes of the hyperbola.

Vertices: $\text{A}=(\text{a},0)\mathrm{\&}{A}^{\prime }=(-\text{a},0)$

Focal Chord: A chord which passes through a focus is called a focal chord.

Double Ordinate: $A$ chord perpendicular to the transverse axis is called a double ordinate.

Latus Rectum (l) L: The focal chord perpendicular to the transverse axis is called the latus rectum.

$\ell ={\displaystyle \frac{2{\text{b}}^2}{\text{a}}}={\displaystyle \frac{{(\text{C}\cdot A)}^2}{\text{TA}}}=2\text{a}\left({\text{e}}^2-1\right)$

Note:

$l(L\text{R})=2\text{e}$ (distance from focus to directrix).

Centre: The point which bisects every chord of the conic drawn through it is called the centre of the conic $C=(0,0)$, the origin is the center of the hyperbola ${\displaystyle \frac{x}{a}}-{\displaystyle \frac{y}{b}}=1$

Since the fundamental equation to the hyperbola only differs from that to the ellipse in having $-{\text{b}}^2$ instead of ${\text{b}}^2$ it will be found that many propositions for the hyperbola are derived from those for the ellipse by simply changing the sign of ${\text{b}}^2$.

Rectangular or Equilateral Hyperbola: 

The particular kind of hyperbola in which the lengths of the transverse and conjugate axis are equal is called an Equilateral Hyperbola. Note that the eccentricity of the rectangular hyperbola is $\sqrt{2}$.

Conjugate Hyperbola:

Two hyperbolas such that transverse & conjugate axes of one hyperbola are respectively the conjugate & the transverse axes of the other are called Conjugate Hyperbolas of each other.

e.g. ${\displaystyle \frac{x^2}{a^2}}-{\displaystyle \frac{y^2}{b^2}}=1\mathrm{\&}{\displaystyle \frac{-x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$ are conjugate hyperbolas of each other.

Note:

(a) If $e_1$ and $e_2$ are the eccentricities of the hyperbola and its conjugate, then $e_1^{-2}+{e_2}^{-2}=1$.

(b) The foci of a hyperbola and its conjugate are concyclic and form the vertices of a square.

(c) Two hyperbolas are said to be similar if they have the same eccentricity:

(d) Two similar hyperbolas are said to be equal if they have same latus rectum.

(e) If a hyperbola is equilateral, then the conjugate hyperbola is also equilateral.

Auxiliary Circle:

A circle drawn with centre C and T.A. as a diameter is called the Auxiliary Circle of the hyperbola. Equation of the auxiliary circle is $x^2+y^2=a^2$

Note from the figure that P and Q are called the "Corresponding Points" on the hyperbola and the auxiliary circle.

In the hyperbola any ordinate of the curve does not meet the circle on $\text{AA}$' as diameter in real points. There is therefore no real eccentric angle as in the case of the ellipse.

Parametric Representation:

The equation $x=a\sec \theta$ & $y=b\tan \theta$ together represents the hyperbola ${\displaystyle \frac{x^2}{a^2}}-{\displaystyle \frac{y^2}{b^2}}=1$ where $\theta$ is a parameter.

Note that if $\text{P}(\theta )=(\text{a}\sec \theta ,\text{b}\tan \theta )$ is on the hyperbola then, $Q(\theta )=($a$\cos \theta$, a $\tan \theta )$ is on the auxiliary circle.

The equation to the chord of the hyperbola joining two points with eccentric angles $\alpha$ and $\beta$ is given by:

${\displaystyle \frac{x}{a}}\cos {\displaystyle \frac{\alpha -\beta }{2}}-{\displaystyle \frac{y}{b}}\sin {\displaystyle \frac{\alpha +\beta }{2}}=\cos {\displaystyle \frac{\alpha +\beta }{2}}$

Position of a Point With Respect to Hyperbola:

The quantity $S_1={\displaystyle \frac{x_1^2}{a^2}}-{\displaystyle \frac{y_1^2}{b^2}}-1$ is positive, zero or negative according as the point $\left(x_1,y_1\right)$ lies inside, on or outside the curve.

Line and a Hyperbola:

The straight line $y=mx+c$ is a secant, a tangent or passes outside the hyperbola ${\displaystyle \frac{x^2}{a^2}}-{\displaystyle \frac{y^2}{b^2}}=1$ according as $c^2>or=or<a^2{m}^2-b^2$, respectively.

Tangents:

(i) Slope Form $:y=mx\pm \sqrt{a^2{m}^2-b^2}$ can be taken as the tangent to the hyperbola ${\displaystyle \frac{x^2}{a^2}}-{\displaystyle \frac{y^2}{b^2}}=1$

(ii) Point Form: Equation of tangent to the hyperbola ${\displaystyle \frac{x^2}{a^2}}-{\displaystyle \frac{y^2}{b^2}}=1$ at the point $\left(x_1{y}_1\right)$ is ${\displaystyle \frac{x{x}_1}{a^2}}={\displaystyle \frac{y{y}_1}{b^2}}=1$

(iii) Parametric Form: Equation of the tangent to the hyperbola ${\displaystyle \frac{x^2}{a^2}}-{\displaystyle \frac{y^2}{b^2}}=1$ at the point $(a\sec \theta ,b\tan \theta )$  is ${\displaystyle \frac{\text{ x}\text{sec}\theta }{\text{a}}}-{\displaystyle \frac{\text{ y}\text{tan}\theta }{\text{b}}}=1$.

Note:
(i) Point of intersection of the tangents at ${\theta }_1\mathrm{\&}{\theta }_2$ is:

$x={\displaystyle \frac{\cos {\displaystyle \frac{{\theta }_1-{\theta }_2}{2}}}{\cos {\displaystyle \frac{{\theta }_1+{\theta }_2}{2}}}},y=b\tan \left({\displaystyle \frac{{\theta }_1+{\theta }_2}{2}}\right)$

(ii) If$$|{\theta }_1+{\theta }_2|=\pi$$, then tangents at these points${\theta }_1\mathrm{\&}{\theta }_2$ are parallel

(iii) There are two parallel tangents having the same slope m.These tangents touches the hyperbola at the extremities of a diameter.

Normals:

(i) The equation of the normal to the hyperbola ${\displaystyle \frac{x^2}{a^2}}-{\displaystyle \frac{y^2}{b^2}}=1$ at the point $P(x_1,y_1)$on it is ${\displaystyle \frac{a^{2}x}{x_1}}+{\displaystyle \frac{b^{2}y}{y_1}}=a^2+b^2=a^2{e}^2$

(ii) The equations of the normal at the point $\text{P}(\text{a}\sec \theta ,\text{b}\tan \theta )$on the hyperbola

${\displaystyle \frac{x^2}{a^2}}-{\displaystyle \frac{y^2}{b^2}}=1\text{ is }{\displaystyle \frac{ax}{\sec \theta }}+{\displaystyle \frac{by}{\tan \theta }}=a^2+b^2=a^2{e}^2$

(iii) Equation of a normal in terms of its slope 'm' is$y=mx-{\displaystyle \frac{\left(a^2+b^2\right)m}{\sqrt{a^2-b^2{m}^2}}}$

Note:
Equation to the chord of contact, polar, chord with a given middle point, pair of tangents from an external point is to be interpreted as in parabola/circle.

Director Circle:

The locus of the intersection point of tangents which are at right angles is known as the Director Circle of the hyperbola. The equation to the director circle is :$x^2+y^2=a^2-b^2$

If $b^2<a^2$ this circle is real

If ${\text{b}}^2={\text{a}}^2$ (rectangular hyperbola) ,the radius of the circle is zero and it reduces to a point circle at the origin. In this case the centre is the only point from which the tangents at right angles can be drawn to the curve. 

If ${\text{b}}^2>{\text{a}}^2$, the radius of the circle is imaginary, so that there is no such circle  & no pair of tangents at right  angle can be drawn to the curve.

Diameter (Not in syllabus):

The locus of the middle points of a system of parallel chords with slope 'm' of an hyperbola is called its diameter. It is a straight line passing through the centre of the hyperbola and has the equation $y=+{\displaystyle \frac{b^2}{a^{2}m}}x$

Note: All diameters of the hyperbola pass through its centre.

Asymptotes (Not in syllabus):

Definition: If the length of the perpendicular let fall from a point on a hyperbola to a straight line ends to zero as the point on the hyperbola moves to infinity along the hyperbola, then the straight line is called the Asymptote of the hyperbola.

Equation of Assymptote : ${\displaystyle \frac{x}{a}}+{\displaystyle \frac{y}{b}}=0$ and ${\displaystyle \frac{x}{a}}-{\displaystyle \frac{y}{b}}=0$

Pairs of Asymptote: ${\displaystyle \frac{x^2}{a^2}}-{\displaystyle \frac{y^2}{b^2}}=0$

Note:

(i) A hyperbola and its conjugate have the same asymptote.

(ii) The equation of the pair of asymptotes different form the equation of hyperbola (or conjugate hyperbola) by the constant term only.

(iii) The asymptotes pass through the centre of the hyperbola and are equally inclined to the transverse axis of the hyperbola. Hence the bisectors of the angles between the asymptotes are the principle axes of the hyperbola.

(iv)The asymptotes of a hyperbola are the diagonals of the rectangle formed by the lines drawn through the extremities of each axis parallel to the other axis.

(v) Asymptotes are the tangent to the hyperbola from the centre.

(vi) A simple method to find the co-ordinates of the centre of the hyperbola expressed as a general equation of degree 2 should be remembered as : Let $f(x,y)=0$ represents a hyperbola.

Find ${\displaystyle \frac{\partial \text{f}}{\partial \text{x}}}$ and ${\displaystyle \frac{\partial \text{f}}{\partial \text{y}}}$. Then the point of intersection of ${\displaystyle \frac{\partial f}{\partial x}}=0$ and ${\displaystyle \frac{\partial f}{\partial y}}=0$ gives the centre of the hyperbola.

Rectangular Hyperbola $(\text{xy}={\text{c}}^2)$:

It is referred to its asymptotes as axes of co-ordinates.

Vertices: $(c,c)$ and $(-c,-c)$

Foci: $(\sqrt{2}c,\sqrt{2}c)\mathrm{\&}(-\sqrt{2}c,-\sqrt{2}c)$

Directrices:$x+y=\pm \sqrt{2}c$

Latus Rectum (l):  $l=2\sqrt{2}c$= T.A. = C.A.

Parametric equation $x=ct,y={\displaystyle \frac{c}{t}},t\in R-\{0\}$

Equation of a chord joining the points $P\left(t_1\right)$& $Q\left(t_2\right)$ is $\text{x + }{\text{t}}_{\text{1}}{\text{t}}_{\text{2}}\text{y = c}\left({\text{t}}_{\text{t}}\text{ + }{\text{t}}_{\text{3}}\right)$

Equation of the tangent at $\text{P}\left({\text{x}}_{\text{l}},y_1\right)$ is ${\displaystyle \frac{\text{x}}{{\text{x}}_1}}+{\displaystyle \frac{\text{y}}{{\text{y}}_1}}=2$ and at $P(t)$ is ${\displaystyle \frac{x}{t}}+ty=2c$