Question

If from any point on the asymptote a straight line be drawn perpendicular to the transverse axis, the product of the segments of this line, intercepted between the point & the curve is always equal to _____

• A. square of the conjugate axis
• B. square of the transverse axis
• C. square of the semi-conjugate axis
• D. square of the semi transverse axis

Answer 1

The correct option is C

square of the semi-conjugate axis

Let's take any point on the asymptotes p,

straight line perpendicular to transverse axis is drawn

which intersect the hyperbola at $Q\&Q^{\prime }$

we need to find $pQ\times p{Q}^{\prime }$

Let take parametric equation on hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

point $Q=(asec\theta ,btan\theta )$

point p on the parabola,

x-coordinate must be the same and equation of asymptotes $y=\frac{b}{a}x$

$y=\frac{b}{a}xsec\theta$

$y=bsec\theta$

Co-ordinate of point $P(asec\theta ,bsec\theta$)

Coordinate of point $Q(asec\theta ,btan\theta$)

by symmetry, we can say point $Q^{\prime }\equiv (asec\theta ,-btan\theta$)

length of PQ

here, x-coordinate is same

$PQ=|b(sec\theta -tan\theta$)|

$P{Q}^{\prime }=|b(sec\theta +tan\theta$)|

$PQ\times P{Q}^{\prime }=b(sec\theta -tan\theta )\times b(sec\theta +tan\theta$)

= $b^2(se{c}^2\theta -ta{n}^2\theta$)

= $b^2\times 1=b^2$

⇒ Product of segment of this line on the curve is equal to the square of the conjugate axis.