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Question

Let F1(x1,0) and F2(x2,0), for x1<0 and x2>0, be the foci of the ellipse x29+y28=1. Suppose a parabola having vertex at the origin and focus at F2 intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant.

If the tangents to the ellipse at M and N meet at R and the normal to tha parabola at M meets the xaxis at Q, then the ratio of area of the triangle MQR to area of the quadrilateral MF1NF2 is

A
3:4
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B
4:5
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C
5:8
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D
2:3
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Solution

The correct option is C 5:8

Equation of tangent at point M (32,6) to the ellipse is x(32)9+y68=1
Put y=0R is (6,0) [R lies on x axis)

Equation of the normal to the parabola at point M(32,6) is
y6=62(x32)
Put y=0Q is (72,0) [Q lies on x axis)

Area of MQR=12×6×52=564

Area of quadrilateral MF1NF2=Area of MF1F2+Area of NF1F2=6+6=26
Area of MQR:Area of quadrilateral MF1NF2=5:8

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