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  • If the normal at any point P on the ellipse cuts the major and minor axes in G and <

If the normal at any point P on the ellipse cuts the major and minor axes in G and g respectively and C be the centre of the ellipse, then 

Option: 1

\mathrm{a^2(C G)^2+b^2(C g)^2=\left(a^2-b^2\right)^2}


Option: 2

\mathrm{a^2(C G)^2-b^2(C g)^2=\left(a^2-b^2\right)^2}


Option: 3

\mathrm{a^2(C G)^2-b^2(C g)^2=\left(a^2+b^2\right)^2}


Option: 4

None of these


Answers (1)

best_answer

Let at point \mathrm{\left(x_1, y_1\right)}  normal will be \mathrm{\frac{\left(x-x_1\right)}{x_1} a^2=\frac{\left(y-y_1\right) b^2}{y_1}}
\mathrm{\text { At } G, y=0 \Rightarrow x=C G=\frac{x_1\left(a^2-b^2\right)}{a^2} \text { and at } g, x=0 \Rightarrow y=C g=\frac{y_1\left(b^2-a^2\right)}{b^2}}\mathrm{\frac{x_1^2}{a^2}+\frac{y_1^2}{b^2}=1 \Rightarrow a^2(C G)^2+b(C g)^2=\left(a^2-b^2\right)^2 .}

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