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

Options

A. `3 : 4`
B. `4:5`
C. `5:8`
D. `2:3`

Answer

C

Explanation

The equation of tangents to the ellipse of M and N are `(x)/(6)+(sqrt6)/(8)y=1 and (x)/(6)-(sqrt6)/(8)y=1` which intersect at R(6, 0).
the equation of the normal to the parabola at M`((4)/(2),sqrt6)` is
`y-sqrt6=-(sqrt6)/(2)(x-(3)/(2))`.
This meets x-axis at `Q((7)/(2),0)`

Now,
`Delta_(1)="Area of " DeltaMQR=(1)/(2)(6-(7)/(2))xxsqrt6=(5sqrt6)/(4)`a
`Delta_(2)="Area of quad. "MF_(1)NF_(2)`
`Delta_(2)=2"Area of "DeltaMF_(1)F_(2)=2((1)/(2)xxF_(1)F_(2)xxsqrt6)=2sqrt6`
`therefore Delta_(1):Delta_(2)=(5sqrt6)/(4):2sqrt6=5:8`