Question

A straight line is drawn from the point $(5,4)$ perpendicular to the transverse axis of the hyperbola
$\frac{x^2}{25}-\frac{y^2}{16}=1$. Then the product of the segments of the line intercepted between the point and the curve is

• A. 16

Answer 1

The correct option is A 16
Given: Hyperbola: $\frac{x^2}{25}-\frac{y^2}{16}=1,$ Point: $(5,4)$

Here $a=5,b=4$
Equation of asymptote of hyperbola $\frac{x^2}{25}-\frac{y^2}{16}=0$

We can see that the point lies on the asymptote of hyperbola.

And we know that, from any point on the asymptotes, a straight line is drawn perpendicular to the transverse axis, then the product of the segments of the line intercepted between the point and the curve is always equal to the square of semi-conjugate axis

$\therefore$ Product of the segments $=b^2=4^2=16$