The locus of a point which moves such that the difference of its distance from two fixed point is always constant is
A). A straight line
B). A circle
C). An ellipse
D). A hyperbola
VerifiedComplete step-by-step solution -
Hyperbola is "the set of all points in a plane such that the difference of the distances from two fixed points (foci) is constant".
The difference of the distances to any point on the hyperbola (x, y) from the two foci (c, 0) and (-c,0) is a constant. That constant will be 2a.
If we let $d_1$ and $d_2$ bet the distances from the foci to the point, then $|d_1−d_2|=2a$.
The absolute value is around the difference so that it is always positive.
We can use that definition to derive the equation of a hyperbola, but I'll give you the short form below.
The only difference in the definition of a hyperbola and that of an ellipse is that the hyperbola is the difference of the distances from the foci that is constant and the ellipse is the sum of the distances from the foci that is constant.
Instead of the equation being $\left(\dfrac{x}{a}\right)^2 + \left(\dfrac{y}{b}\right)^2=1$, the equation is $\left(\dfrac{x}{a}\right)^2 + \left(\dfrac{y}{b}\right)^2=1$
The graphs, however, are very different.
The center is the starting point at (h, k).
The Transverse axis contains the foci and the vertices.
Transverse axis length = 2a. This is also the constant that the difference of the distances must be.
Conjugate axis length = 2b.
Distance between foci = 2c.
The foci are within the curve.
Since the foci are the farthest away from the center, c is the largest of the three lengths, and the Pythagorean relationship is: $a^2+b^2=c^2$
Hence the correct option is D.
Note- In order to solve these types of questions, you need to remember the basic definition of locus of all shapes such as for circle which is a locus of collection of points which are equidistant from the center point. Similarly for parabola in which every point is equidistant from a point called focus and a straight line called directrix.
