Ellipse
Geometry of Parabola and Ellipse
JEE Advanced 2016 Paper 2
Grade 11
Question:
Let $F_1(x_1, 0)$ and $F_2(x_2, 0)$ with $x_1 < 0 < x_2$ be the foci of the ellipse $\frac{x^2}{9} + \frac{y^2}{8} = 1$. A parabola with vertex at the origin and focus at $F_2$ intersects the ellipse at $M$ (first quadrant) and $N$ (fourth quadrant). If the tangents to the ellipse at $M$ and $N$ meet at $R$, and the normal to the parabola at $M$ meets the $x$-axis at $Q$, find the ratio of area of $\triangle MQR$ to area of quadrilateral $MF_1NF_2$. _____
Step-by-Step Solution
Key Concept: Ellipse: $a^2 = 9, b^2 = 8, c = 1$; $F_2 = (1, 0)$. Parabola: $y^2 = 4x$. Intersection: $x^2/9 + (4x)/8 = 1 \Rightarrow x^2/9 + x/2 = 1 \Rightarrow 2x^2 + 9x - 18 = 0 \Rightarrow x = 3/2, y = \pm \sqrt{6}$. $M = (3/2, \sqrt{6})$.
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Correct Answer: \frac{3}{4}