Question

The locus of middle points of normal chords of $x^2-y^2=a^2$ is :

• A. $(y^2-x^2{)}^3=4a^2{x}^2{y}^2$
• B. $(y^2-x^2{)}^4=4a^4{x}^2{y}^2$
• C. $(y^2+x^2{)}^3=2a^4{x}^2{y}^2$
• D. $(y^2+x^2{)}^3=2a^2{x}^2{y}^2$

Answer 1

Answer: (A)

$(y^2-x^2{)}^3=4a^2{x}^2{y}^2$

$ax\cos \theta +ay\cot \theta =2a^2$ ($e{q}^n$ of normal)
$x\cos \theta +y\cot \theta =2a$-----(1)
$x=\left(\frac{2a-y\cot \theta }{\cos \theta }\right)$
$y^2({\cot }^2\theta -{\cos }^2\theta )-4ay\cot \theta +(4a^2-a^2{\cos }^2\theta )=0$
$y_1+y_2=\frac{4a\cot \theta }{\left({\cot }^2\theta {\cos }^2\theta \right)}$------(2)
$x^2({\cot }^2\theta -{\cos }^2\theta )+4ax\cos \theta -4a^2-a^2{\cot }^2=0$
$x_1+x_2=\frac{-4a\cos \theta }{({\cot }^2\theta -{\cos }^2\theta )}$-----(3)
$h=\frac{x_1+x_2}{2},k=\left(\frac{y_1+y_2}{2}\right)$-----(4)
By solving (1),(2),(3) and (4) and eliminating $\theta$ we get
$(k^2-h^2{)}^3=4a^2{k}^2{h}^2$
$\therefore (y^2-x^2{)}^3=4a^2{x}^2{y}^2$