Question

A normal to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ meets the transverse and conjugate axes in M and N and the lines MP and NP are drawn at right angle to the axes. The locus of P is

• A. The parabola
• B. The circle
• C. The ellipse
• D. The hyperbola

Answer 1

The correct option is D The hyperbola

Normal to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ at the point $Q(asec\varphi ,btan\varphi )$ is
ax cos $\varphi$ + by $cot\varphi =a^2+b^2$
Let coordinates of P be (h, k), then
$h=\frac{(a^2+b^2)}{a}sec\varphi andk=\frac{(a^2+b^2)}{b}tan\varphi$
$\Rightarrow$ $ah=(a^2+b^2)sec\varphi$ …… (i)
and $bk=(a^2+b^2)tan\varphi$ …… (ii)
On squaring and subtracting equation (ii) from equation (i), we get $a^2{h}^2-b^2{k}^2=(a^2+b^2{)}^2$
Then, locus of P is
$a^2{x}^2-b^2{y}^2=(a^2+b^2{)}^2$