Question

Locus of the mid points of the chord of ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$, so that chord is always touching the circle $x^{2} + y^{2} = c^{2},$

(c < a, c < b) is

• A. $\left( b^{2}x^{2} + a^{2}y^{2} \right)^{2} = c^{2}\left( b^{4}x^{2} + a^{4}y^{2} \right)$
• B. $\left( a^{2}x^{2} + b^{2}y^{2} \right)^{2} = c^{2}\left( a^{4}x^{2} + b^{4}y^{2} \right)$
• C. $\left( a^{2}x^{2} + b^{2}y^{2} \right)^{2} = c^{2}\left( a^{4}x^{2} + b^{4}y^{2} \right)$
• D. None of these

Answer 1

Answer: (A)

$\left( b^{2}x^{2} + a^{2}y^{2} \right)^{2} = c^{2}\left( b^{4}x^{2} + a^{4}y^{2} \right)$

Answer 2

Let mid point of the chord be (h, k), then equation of the chord is $\frac{hx}{a^{2}} + \frac{ky^{2}}{b^{2}} - 1 = \frac{h^{2}}{a^{2}} + \frac{k^{2}}{b^{2}} - 1$⇒ y =

$\frac{b^{2}}{a^{2}}.\frac{h}{k}x + \left( \frac{h^{2}}{a^{2}} + \frac{k^{2}}{b^{2}} \right)\frac{b^{2}}{k}$ ---- 1

since line (1) is touching the circle x² +y² = c² ---- 2

∴ $\left( \frac{h^{2}}{a^{2}} + \frac{k^{2}}{b^{2}} \right)^{2}\frac{b^{4}}{k^{2}} = c^{2}\left( 1 + \frac{b^{4}}{a^{4}}\frac{h^{2}}{k^{2}} \right)$

∴ Required locus is $\left( b^{2}x^{2} + a^{2}y^{2} \right)^{2} = c^{2}\left( b^{4}x^{2} + a^{4}y^{2} \right)$.