Question

If the locus of the middle point of the chords of contact of tangents to $x^2-y^2=a^2$ from points on the auxiliary circle is of the form $p{x}^4+q{y}^2{x}^2+p{y}^4-r(x^2+y^2)=0$ then $r=$

• A. $a^2$
• B. $-a^2$
• C. $a$
• D. $-a$

Answer 1

Answer: (A)

$a^2$

Any point on the auxiliary circle $x^2+y^2=a^2$ is $P(a\cos \theta ,a\sin \theta )$ chord of contact of tangents from $P(a\cos \theta ,a\sin \theta )$ to $x^2-y^2=a^2$ is
$x\cos \theta -y\sin \theta =a.....\left(1\right)$
If $(x_1,y_1)$ is the mid-point of this chord, then its equation is
$T=S_1(i.e.,)x{x}_1-y{y}_1=x_1^2-y_1^2....\left(2\right)$
Identifying (1) and (2) we have $\frac{\cos \theta }{x_1}=\frac{\sin \theta }{y_1}=\frac{a}{x_1^2-y_1^2}$
Eliminating $\theta$ ,we have $1=\frac{a^2{x}_1^2+a^2{y}_1^2}{(x_1^2-y_1^2{)}^2}$
$\therefore$ locus of $(x_1,y_1)is(x^2-y^2{)}^2-a^2(x^2+y^2)=0$
Hence $r=a^2$