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  • In an ellipse the perpendicular from a focus upon any tangent and the line joining the centre of the ellipse to the point of contact meet on the:Option: 1</stron

In an ellipse the perpendicular from a focus upon any tangent and the line joining the centre of the ellipse to the point of contact meet on the:

Option: 1

Corresponding tangent at vertex
 


Option: 2

corresponding directrix
 


Option: 3

non -corresponding tangent at vertex
 


Option: 4

non-corresponding directrix


Answers (1)

best_answer

Let the ellipse be \mathrm{\frac{x^2}{a^2}+\frac{y^2}{b^2}=1}and O be the centre.

Tangent at \mathrm{P\left(x_1, y_1\right)} is \mathrm{\frac{x x_1}{a^2}+\frac{y y_1}{b^2}-1=0 \quad} whose slope \mathrm{=-\frac{b^2 x_1}{a^2 y_1}}

Focus is \mathrm{\mathrm{S}(\mathrm{ae}, 0)}

Equation of the line perpendicular to tangent at P through S is

\mathrm{ \begin{aligned} & y=\frac{a^2 y_1}{b^2 x_1}(x-a e) \ldots(1) \\ & \text { Equation of } O P \text { is } y=\frac{y_1}{x_1} x \end{aligned} .....(2)}
(1) and (2) intersect

\mathrm{ \Rightarrow \quad \frac{y_1}{x_1} x=\frac{a^2 y_1}{b^2 x_1}(x-a e) \Rightarrow x\left(a^2-b^2\right)=a^3 e \Rightarrow x \cdot a^2 e^2=a^3 e \Rightarrow x=a / e }

which is the corresponding directrix.

Posted by

Ritika Harsh

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