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Question

Let PQ be a focal chord of y2=4ax. The tangents to parabola at P and Q meet at a point lying on the line y=2x+a (a>0). If the chord PQ subtends an angle θ at the vertex of prabola then tanθ=

A
273
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B
275
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C
253
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D
235
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Solution

The correct option is C 253
Let P(at21,2at1) and Q(at22,2at2)
For focal chord t1t2=1
Let O be the vertex, then slope of PO=m1=2t1
And slope of QO=m2=2t2
tanθ=m1m21+m1m2=2(t2t1)t1t2+4
Let point of intersection of tangents at t1,t2 be R=(at1t2,a(t1+t2))
R lies on the directix x=a and it also lies on line y=2x+a
y=a=a(t1+t2)
t1+t2=1
(t2t1)2=(t1+t2)24t1t2=5
t2t1=±5
tanθ=251+4=253

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