Question

PN is the ordinate of any point P on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If Q divides AP in the ratio $a^2:b^2$ then NQ is

• A. perpendicular to A’P
• B. parallel to A’P
• C. perpendicular to OP
• D. none

Answer 1

The correct option is A perpendicular to A’P

$A=(a,0);A^{\prime }=(-a,0);P(asec\theta ,btan\theta )andN=(asec\theta ,0)$; the point Q which divides AP in the ratio
$a^2:b^2$ is $Q=(\frac{a^{3}sec\theta +a{b}^2}{a^2+b^2},\frac{a{b}^{2}tan\theta }{a^2+b^2})$
$\therefore$ the product of slope of NQ and slope of A’P is - 1.