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Question:

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, then the ratio of area of the triangle MQR to area of the quadrilateral MF1NF2 is 3:4, 4:5, 5:8, or 2:3?

3:4

4:5

5:8

2:3

Solution:

Equation of tangent at M is: x/32 + y√68 = 1
Let us put y=0 as the intersection will be on X-axis.
∴R=(6,0)
Equation of normal at m is: √32x + y = 2√32 + (√32)³
Putting y=0, we get: x = 2 + 32 = 72
∴Q=(72,0)
∴Area (ΔMQR) = 1/2 × (6 - 0) × √6 = 54√6 sq. unit
Area of quadrilateral (MF1NF2) = 2 × Area (ΔF1F2M) = 2 × 1/2 × 2 × √6 = 2√6 sq. unit
∴required ratio = 54/2 = 58