AB is a double ordinate of the parabola y 2 = 4ax. Tangents drawn to parabola at A and B meets y-axis at A 1 and B 1 respectively. If the area of trapezium AA 1 B 1 B is equal to 12a 2 , then angle subtended by A 1 B 1 at the focus of the parabola is equal to -
Text Solution
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Let A ≡ (at 1 2 , 2at 1 ), B ≡ (at 1 2 , –2at 1 ). Equation of tangents at A and B are
yt 1 = x + at 1 2 and yt 2 = x + at 2 2 , respectively.
A 1 ≡ (0, at 1 ), B 1 ≡ (0, –at 2 )
Area of trapezium AA 1 B 1 B
= 
(AB + A 1 B 1 ) . OC
⇒ 24a 2 = 
. (4at 1 + 2at 1 ) (at 1 2 )
⇒ t 1 3 = 8 ⇒ t 1 = 2 ⇒ A 1 ≡ (0, 2a)
If ∠ OSA 1 = θ
⇒ tan θ = 
= 2 ⇒ θ = tan –1 (2).
Thus, required angle is 2tan –1 (2).
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