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A B is a double ordinate of the parabola...

`A B` is a double ordinate of the parabola `y^2=4a xdot` Tangents drawn to the parabola at `Aa n dB` meet the y-axis at `A_1a n dB_1` , respectively. If the area of trapezium `AA_1B_1B` is equal to `12 a^2,` then the angle subtended by `A_1B_1` at the focus of the parabola is equal to `2tan^(-1)(3)` (b) `tan^(-1)(3)` `2tan^(-1)(2)` (d) `tan^(-1)(2)`

A

`pi/2`

B

`tan^(-1)(3/4)`

C

`tan^(-1) ((-4)/(3))`

D

`pi/3`

Text Solution

Verified by Experts

The correct Answer is:
C
`p(at_(1)^(2),2at_(1))" "t_(2)=-t_(1)`
`Q(at_(2)^(2),2at_(2))`
Equation of tangent at P `" "t_(1)y=x+at_(1)^(2)`
`S-=(0, at_(1))`
`R-=(0, -at_(1))`

Area of trapezium PQRS
`=(1)/(2)(2at_(1)+4at_(1))xxat_(1)^(2)=24a%^(2)`
`=3a^(2)t_(1)^(3)=24a^(2)`
`t_(1)^(3)=(24)/(3)=8`
`t_(1)=2`
`tan theta=(at_(1))/(a)=t_(1)=2`
Angle subtended by SR at focus
`tan 2 theta =(2 tan theta)/(1-tan^(2)theta)=(4)/(1-4)=(-4)/(3), alpha=tan^(-1)((-4)/(3))`
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