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If the tangents to the ellipse at M and ...

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

A

`3:4`

B

`4:5`

C

`5:8`

D

`2:3`

Text Solution

Verified by Experts

Equation of tangent to ellipse at M is : `(3x)/(2xx9)+(sqrt(6)y)/(8)=1`
Put y=0 as intersection will be on x-axis
`:. R-=(6,0)`
Equation of normal to parabola, at M is `sqrt((3)/(2))x+y=2sqrt((3)/(2))+(sqrt((3)/(2)))^(3)`
Put `y=0,2+(3)/(2)=(7)/(2)`
`.:. Q-=((7)/(2),0)`
`:. "Area" (DeltMQR)=(1)/(2)xx(1-(7)/(2))xxsqrt(6)=(5)/(4)` sq. units
Area of quadrilateral `(MF_(1)NF_(2))`
`2xx"Area"(DeltaF_(1)F_(2)M)`
`=2xx(1)/(4)xx2xxsqrt(6)=2sqrt(6)` sq. units
`:.` Required Ratio `=(4)/(2)=(5)/(8)`
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