Question

Consider the hyperbola xy = 4 and a line 2x + y = 4. O is the centre of the hyperbola. Tangent at any point P on the hyperbola intersects the coordinate axes at A and B.

Let the given line intersect x-axis at R. If a line through R intersects the hyperbola at S and T, then the minimum value of (RS) (RT) is ___

• A. $4$
• B. $16$
• C. $8$
• D. $4\sqrt{2}$

Answer 1

The correct option is C

$8$

Locus of circum centre is rectangular hyperbola.
R = (2, 0) Any point through R is $(2+rcos\theta ,rsin\theta )$
$\Rightarrow (2+rcos\theta )\left(rsin\theta \right)=4\Rightarrow r^{2}sin\theta cos\theta +2rsin\theta =4\therefore r_1{r}_2=\left|\frac{4}{sin\theta cos\theta }\right|\ge 8$