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

Answer 1

Answer: (A)

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

Answer 2

The given hyperbola is $\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1$ ……(i)

Any tangent to (i) is $y = mx + \sqrt{16m^{2} - 9}$

……(ii)

Let $(x_{1},y_{1})$ be the mid point of the chord of the circle $x^{2} + y^{2} = 16$

Then equation of the chord is $T = S_{1}$ i.e.,

$xx_{1} + yy_{1} - (x_{1}^{2} + y_{1}^{2}) = 0$ ……(iii)

Since (ii) and (iii) represent the same line.

$\therefore$ $\frac{m}{x_{1}} = \frac{- 1}{y_{1}} = \frac{\sqrt{16m^{2} - 9}}{- (x_{1}^{2} + y_{1}^{2})}$

⇒ $m = - \frac{x_{1}}{y_{1}}$ and $(x_{1}^{2} + y_{1}^{2})^{2} = y_{1}^{2}(16m^{2} - 9)$

⇒$(x_{1}^{2} + y_{1}^{2})^{2} = 16.\frac{x_{1}^{2}}{y_{1}^{2}}y_{1}^{2} - 9y_{1}^{2}$ = $16x_{1}^{2} - 9y_{1}^{2}$

$\therefore$ Locus of $(x_{1},y_{1})$ is $(x^{2} + y^{2})^{2} = 16x^{2} - 9y^{2}$.