The centre of a circle C is at the centre of the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a \gt b$. Let $C$ pass through the foci $F_1$ and $F_2$ of $E$ such that the circle $C$ and the ellipse $E$ intersect at four points. Let P be one of these four points. If the area of the triangle…

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The centre of a circle C is at the centre of the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a \gt b$. Let $C$ pass through the foci $F_1$ and $F_2$ of $E$ such that the circle $C$ and the ellipse $E$ intersect at four points. Let P be one of these four points. If the area of the triangle $\mathrm{PF}_1 \mathrm{~F}_2$ is 30 and the length of the major axis of E is 17 , then the distance between the foci of E is :

Aicharya · Accepted Answer

$\begin{aligned} & \frac{1}{2} \mathrm{PF}_1 \cdot \mathrm{PF}_2=30 \\ & \mathrm{PF}_1+\mathrm{PF}_2=17 \\ & \mathrm{PF}_1=12 \mathrm{PF}_2=5 \\ & \mathrm{~F}_1 \mathrm{~F}_2=13 \\ & \text { option }(2)\end{aligned}$

The centre of a circle C is at the centre of the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a \gt b$.…

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