Latus Rectum

A conic section is defined in mathematics as a curve formed by the intersection of the surface of a cone with a plane. Conic sections are classified into three categories. They are the parabola, the ellipse, and the hyperbola. Many key terminologies are employed to characterise these curves, such as focus, directrix, latus rectum, locus, asymptote, and so on. Let us go through the phrase "Latus Rectum" in depth in this post.

A conic section is defined by a second-degree polynomial equation in two variables. Conic sections are classified into three different types namely ellipse, parabola, and hyperbola. The different names are given to the conic section as each conic section is represented by a cross-section of a plane cutting through a cone. To define these curves, many important terms are used such as latus rectum, focus, directrix, etc. In this article, we will discuss the latus rectum of different curves such as a parabola, hyperbola, and ellipse in detail.

The chord across the focus and parallel to the directrix in the conic section is the latus rectum. The term latus rectum is derived from the Latin words "latus" (side) and "rectum" (straight). The semi latus rectum is half of the latus rectum. The figure above depicts a parabola's latus rectum.

What is Latus Rectum?

In the conic section, the word ‘latus rectum’ is derived from the Latin word ‘latus’ which means ‘side’ and the rectum which means ‘straight’. The latus rectum is defined as the chord passing through the focus, and perpendicular to the directrix. The end point of the latus rectum lies on the curve. Half of the latus rectum is considered as the semi latus rectum. The length of the latus rectum of each conic section is defined differently. For example,

• The length of the latus rectum of the parabola is always equivalent to four times the focal length of the parabola.

• The length of the latus rectum of an ellipse is defined as the square of the length of the conjugate axis divided by the length of the transverse axis.

• The length of the latus rectum of a hyperbola is defined as the square of the length of the transverse axis divided by the length of the conjugate axis.

• The length of the latus rectum of a circle is equivalent to the length of its diameter.

Parabola

A parabola is defined in terms of a line, known as the directrix, and a point not on the directrix is considered as the locus of points that are equidistant from both the directrix and focus. The axis of symmetry is a line that splits the parabola symmetrically. The vertex of the parabola is the intersection of the axis of symmetry and the parabola.

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Latus Rectum of Parabola

The latus rectum of the parabola is a line segment passing through its focus and perpendicular to its axis. The length of the latus rectum of a parabola is always equivalent to four times the distance of the focus from the vertex of the parabola. 

General Equation of Parabola

Let us consider the situation where the axis of the parabola is perpendicular to the y-axis. Accordingly, its equation will be of the type (x - h) = 4a(y-k), where the variables h, a, and k are considered as the real numbers, ( h, k) is its vertex, and 4a is the latus rectum

If we rearrange the formula, we get

x² - 2hx + h² = 4ay - 4ak

4ay = x² - 2hx + h² + 4ak

y = 1/4a ( x² - 2hx + h² + 4ak)

= 1/4a x²  + (- 2h/4a x ) + ((h2 + 4ak)/ 4a)

If we replace 1/4a with p,  -2h/4a with q , and h² + 4ak/4a with r , we get the quadratic equation as

y = px² + qx + r

In the above quadratic equation, p cannot be 0, or else the equation will have a straight line.

Length of the Latus Rectum of Parabola Derivation

Let the end of the latus rectum of a parabola y = 4ax as L and L’. The x - coordinates of L and L’ are equivalent to ‘a’ as S =( a, 0)

Let us consider L = (a,b)

As we know L is  a point of parabola, we have

b² = 4a (a) = 4a²

Taking square root on both the sides, we get b = 2a

Hence, the end of the latus rectum of parabola are L = ( a, 2a) and L’ =( a, -2a)

Therefore, the length of the latus rectum of parabola LL’ is 4a.

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Latus Rectum of Parabola Formula

The Latus rectum of the parabola formula is 4a, where a is the distance of the focus from the vertex of the parabola.

Hyperbola

A hyperbola is defined as the locus of a point in such a way that the distance to each focus is greater than 1. In other words, the locus of points moving in a plane in such a way that the ratio of its distance from a fixed point i.e. (focus) to its distance from a fixed line ( directrix) is constant and greater than 1.

Note: The point (focus) does not lie on the line (directrix) i.e. (PS/PM) = e > 1 eccentricity.

Latus Rectum of Hyperbola 

The Latus rectum of a hyperbola is defined as a line segment perpendicular to the transverse axis through any of the foci and whose ending point lies on the hyperbola. The length of the latus rectum of a hyperbola is 2b²/a.

Ellipse

The set of all points in a plane, the sum of the distance from the fixed point in the plane is constant, is an ellipse. The two fixed points in the plane are considered as the foci of the ellipse. When a line segment is formed joining two focus points, the midpoint of this line is considered as the centre of an ellipse.

The line joining two foci is known as the major axis and a line forms through the centre and perpendicular to the major axis is the minor axis. The ending points of the major axis are known as the vertices of the ellipse.

An ellipse is the locus of all the points on a plane whose distances from two fixed points in the plane are constant. The fixed points, which are encompassed by the curve, are known as foci (singular focus). The constant ratio is the eccentricity of the ellipse, and the fixed line is the directrix. Eccentricity is a factor of the ellipse that shows its elongation and is symbolised by the letter 'e.'

The ellipse has an oval form, and the area of an ellipse is specified by its major and minor axes. Ellipse area = ab, where a and b are the lengths of an ellipse's semi-major and semi-minor axes. Ellipse is analogous to other portions of the conic section that are open and unbounded in shape, such as parabola and hyperbola.

Definition

In terms of locus, an ellipse is the set of all points on an XY plane whose distance from two fixed points (called foci) adds up to a constant number.

The ellipse is a kind of conic section formed when a plane cuts a cone at an angle with its base. If the plane intersects the cone parallel to the base, it produces a circle.

Shape of an Ellipse

An ellipse is a two-dimensional form specified along its axes in geometry. When a cone is crossed by a plane at an angle with respect to its base, an ellipse is created.

It has two main sites of interest. For all points in a curve, the total of the two distances to the focal point is always constant.

Major Axis and Minor Axis

Ellipses are characterised by their two axes along the x and y axes:

• Main axis

• Minor Axis

The main axis is the ellipse's longest diameter (typically represented as 'a'), which runs through the centre from one end to the other, at the broadest section of the ellipse. The minor axis is the ellipse's smallest diameter (denoted by 'b'), passing through the centre at its narrowest point.

Half of the major axis is known as the semi-major axis, while half of the minor axis is known as the semi-minor axis.

Properties

• Ellipse has two focal points, which are also known as foci.

• A set distance is referred to as a directrix.

• The eccentricity of an ellipse ranges from 0 to 1.

The latus rectum is a line that runs parallel to the conic's directrix and passes through its foci. The focal chord is the Latus rectum, and the number of latus rectums equals the number of foci in the conic. A parabola has one latus rectum, while an ellipse and hyperbola have two.

Also, 

• The length of the major axis of an ellipse is represented by 2a.

• The length of the minor axis of an ellipse is represented by 2b.

• The distance of the foci is given by 2c

Latus Rectum Examples

1. Find the equation of the parabola having its focus ( 0, -3) and the directrix of the parabola is on the line y = 3

Solution: 

As the focus of the parabola is on the y- axis and is also below the directrix, the parabola will be opened downward, and the value of a = -3. Hence, the equation of a parabola is given as x² = 12x. The length of the latus rectum of a parabola is |4 (-3)| = 12.