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 $4 x=3$
