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The locus of the middle points of the chords of the ellipse $x^{2} / a^{2} + y^{2} / b^{2} = 1$ touching the ellipse $x^{2} / α^{2} + y^{2} / β^{2} = 1 is$

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$\frac{a^{2}}{x^{2}} + \frac{b^{2}}{y^{2}} = a^{2} + b^{2}$

$\frac{x}{a} + \frac{y}{b} = 1$

$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 0$

$\frac{α^{2} x^{2}}{a^{4}} + \frac{β^{2} y^{2}}{b^{4}} = {(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}})}^{2}$

answer is D.

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Detailed Solution

let P(x,y) be locus of midpoint of chord

equation of chord of first ellipse is a tangent to $\frac{x^{2}}{α^{2}} + \frac{y^{2}}{β^{2}} = 1$

$∴ A^{2} I^{2} + B^{2} M^{2} = N^{2}$

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