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  • If S K be the perpendicular from the focus S on the tangent at any point P on the ellipse <img alt="\mathrm{\frac{x^2}{a^2}+\frac{y^2}{b^2}=1}" src="https://entrancecorner.oncodecogs.com/gif.latex?

If S K be the perpendicular from the focus S on the tangent at any point P on the ellipse \mathrm{\frac{x^2}{a^2}+\frac{y^2}{b^2}=1} , then locus of K is

 

Option: 1

\mathrm{ a^2 x^2+b^2 y^2=(a x-b y)^2}


Option: 2

\mathrm{ x^2+y^2=a^2}


Option: 3

\mathrm{ x^2+y^2=b^2}


Option: 4

\mathrm{ x^2+y^2=a^2+b^2}


Answers (1)

Tangent at any point \mathrm{(a \cos \theta, b \sin \theta)} on the ellipse will be

\mathrm{\frac{x \cos \theta}{a}+\frac{y \sin \theta}{b}=1\quad \quad \quad \quad \dots(i)}

A line perpendicular to (i) and passing through focus (ae, 0) will be

\mathrm{\frac{x \sin \theta}{b}-\frac{y \cos \theta}{a}=\frac{a \theta \sin \theta}{b} \quad \quad \quad \quad \dots(ii)}

By eliminating \theta from both the equations, we get the locus of K.

Hence (b) is the correct answer.

Posted by

Kshitij

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