Question

Equation of chord having mid-point (h, k) oblique hyperbola $xy=c^2$ is same as equation of tangent at (h, k) on the hyperbola $xy=c^2$

• A. True
• B. False

Answer 1

The correct option is A

True

we have already learned the equation of chord with a given middle point as

$T=S_1$

Here ,$T=\frac{x}{x_1}+\frac{y}{y_1}-2=0$

For point (h,k)

T is $\frac{x}{h}+\frac{y}{k}-2=0$

$S_1=x_1{y}_1-c^2\equiv hk-c^2$

Equation of the chord with given middle - point is

$T=S_1$

$\frac{x}{h}+\frac{y}{k}-2=hk-c^2$

$kx+yh-2hk=h^2{k}^2-hk{c}^2$ - - - - - - (1)

(h,k) should also satisfy the hyperbola $xy-c^2$

$hk=c^2$

from equation (1)

$ky+yh-2hk=c^4-c^2.c^2=0$

$kx+yh=2hk$

Dividing both sides by hk

we get,

$\frac{y}{h}+\frac{h}{k}=2$

This equation is same as equation of tangent at (h.k) on the hyperbola $xy=c^2$

so,given statement is true