The portion of the tangent intercepted between the point of contact and the directrix of the parabola \( y^2 = 4ax\) subtends at the focus an angle of
A
30∘
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B
45∘
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C
60∘
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D
90∘
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Solution
The correct option is D90∘ If P=(at2,2at) then the equation of tangent at t is ty=x+at2.......(1)
Directrix is x+a=0............(2)
By solving Q=[−a,a(t2−1)t],S=(a,0)
Slope of SP=2ata(t2−1)=2tt2−1,Slope of SQ=a(t2−1)t−2a=−(t2−1)2t
Product of slopes = –1. ∴∠PSQ=90∘