Q116 / 170 · Manipal (MET) 2016 · Mathematics

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

  • A
    $\left(x^2+y^2\right)^2=16 x^2+9 y^2$
  • B
    $\left(x^2-y^2\right)^2=16 x^2-9 y^2$
  • Correct answer
    C
    $\left(x^2+y^2\right)^2=16 x^2-9 y^2$
  • D
    None of the above

Solution

Let $P\left(x_1, y_1\right)$ be the mid-point of a chord of the circle. The equation of the chord is $T=S_1$ i.e. $\quad x x_1+y y_1-16=x_1^2+y_1^2-16$ or $y=-\frac{x_1}{y_1} x+\frac{x_1^2+y_1^2}{y_1}$ Since, it touches the hyperbola $\frac{x^2}{4^2}-\frac{y^2}{3^2}=1$. $\therefore\left(\frac{x_1^2+y_1^2}{y_1}\right)=4^2 \cdot \frac{x_1^2}{y_1^2}-3^2\left[\because c^2=a^2 m^2-b^2\right]$ $\therefore$ Locus of $\left(x_1, y_1\right)$ is $\frac{\left(x^2+y^2\right)^2}{y^2}=\frac{16 x^2-9 y^2}{y^2}$ or $\quad\left(x^2+y^2\right)^2=16 x^2-9 y^2$

Hence, the correct answer is option: 'C'.

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