Name: Equation of Chord of Hyperbola in Parametric Form
Uploaded: 2019-07-19
Duration: 1 min 30 s
Description: How to find the equation of chord of a hyperbola joining two points whose parametric coordinates are known? Check out this video to find out.

Question

A tangent to the parabola

x^{2} = 4 a y

meets the hyperbola

x y = k^{2}

in two points

P

and

Q

. Prove that the middle point of

P Q

lies on the parabola.

Accepted Answer

The equation of the hyperbola is given as,

$x y = k^{2} \dots (1)$

And, the equation of the parabola is given as,

$x^{2} = 4 a y \dots (2)$

If $(α, β)$ are the coordinates of the middle point of some chord of $(1)$ , then the equation to the chord will be given as,

$x α + y β = 2 α β \dots (3)$

If $(3)$ is a tangent to $(2)$ , then on solving the two equations, we must get two coincident values.

Hence, substituting the value of $y$ from equation $(3)$ in equation $(2)$ , we get,

$x^{2} = 4 a (\frac{2 α β - x α}{β})$

$β x^{2} + 4 a α x - 8 a α β = 0 . . . (4)$

Equation $P$ has equal roots if its discriminant is zero, when,

$16 a^{2} α^{2} + 32 a α β^{2} = 0$

Hence, we get,

$a α + 2 β^{2} = 0$

Generalising the above equation, we get the required locus of the midpoint as,

$2 y^{2} + a x = 0$ (which is a parabola)

A tangent to the parabola $x^{2} = 4 a y$ meets the hyperbola $x y = k^{2}$ in two points $P$ and $Q$ . Prove that the middle point of $P Q$ lies on the parabola.

Important Questions on Hyperbola

Learn and Prepare for any exam you want !

A tangent to the parabola x2=4ay meets the hyperbola xy=k2 in two points P and Q. Prove that the middle point of PQ lies on the parabola.

Important Questions on Hyperbola

Learn and Prepare for any exam you want !

A tangent to the parabola $x^{2} = 4 a y$ meets the hyperbola $x y = k^{2}$ in two points $P$ and $Q$ . Prove that the middle point of $P Q$ lies on the parabola.