Question

The locus of the mid points of the chords of the circle $x^2+y^2=16$ which are tangent to the hyperbola $9x^2-16y^2=144$ is :

• A. ${(x^2+y^2)}^2=16x^2+9y^2$
• B. ${(x^2-y^2)}^2=16x^2-9y^2$
• C. ${(x^2+y^2)}^2=16x^2-9y^2$
• D. None of these

Answer 1

The correct option is C ${(x^2+y^2)}^2=16x^2-9y^2$

Let $P(x_1,y_1)$ be the mid point of a chord of the circle.

The equation of the chord is $T=S_1$

$\Rightarrow x{x}_1+y{y}_1-16={x_1}^2+{y_1}^2-16\Rightarrow y=-\frac{x_1}{y_1}x+\frac{{x_1}^2+{y_1}^2}{y_1}$

Since it touches the hyperbola $\frac{x^2}{4^2}-\frac{y^2}{3^2}=1$

$\therefore {\left(\frac{{x_1}^2+{y_1}^2}{y_1}\right)}^2=4^2.\frac{{x_1}^2}{{y_1}^2}-3^2[\because c^2=a^2{m}^2-b^2]$

$\therefore$ Locus of $(x_1,y_1)$ is

$\frac{{(x^2+y^2)}^2}{y^2}=\frac{16x^2-9y^2}{y^2}\Rightarrow {(x^2+y^2)}^2=16x^2-9y^2$