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Question

A tangent to E1:x2+4y2=4 meets E2:x2+2y2=6 at P and Q. Tangents at P and Q of E2 make an angle πn (nN). Then n2=

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Solution

Let any point on x24+y21=1 be (2cosθ,sinθ).
tangent at that point is xcosθ2+ysinθ=1 (1)

Let point of intersection of tangents at P
and Q be R(h,k)
Chord of contact for E2:hx6+ky3=1 (2)
(1) and (2) are indentical
cosθ2h6=sinθk3=1
cosθ=h3,sinθ=k3
h2+k2=9
x2+y2=6+3
Clearly, locus of point of intersection of tangents lie on director circle of E2
Angle between tangents =π2
n2=22=4

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