### Hyperbola Question 1

#### Question 1 - 24 January - Shift 1

Let a tangent to the curve $y^{2}=24 x$ meet the curve $x y=2$ at the points $A$ and $B$. Then the mid points of such line segments $A B$ lie on a parabola with the

(1) directrix $4 x=3$

(2) directrix $4 x=-3$

(3) Length of latus rectum $\frac{3}{2}$

(4) Length of latus rectum 2

## Show Answer

#### Answer: (1)

#### Solution:

#### Formula: Drectrix of hyperbola, Tangent of a curve

$y^{2}=24 x$

$a=6$

$xy=2$

$AB \equiv ty=x+6 t^{2}$

$AB \equiv T=S_1$

$kx+hy=2 hk$

From (1) and (2)

$\frac{k}{1}=\frac{h}{-t}=\frac{2 hk}{-6 t^{2}}$

$\Rightarrow$ then locus is $y^{2}=-3 x$

Therefore directrix is $4 x=3$