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Question

A tangent is drawn to the ellipse E1:9x2+y2=36 to cut the ellipse E2:3x2+y2=48 at the points A and B. If the tangents at A and B to the ellipse E2 intersect at C(p,3), p>0, then the value of [p] is ([.] represents the greatest integer function)

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Solution

E1:x24+y236=1
Equation of any tangent AB to the ellipse E1 is 2xcosθ4+6ysinθ36=1 (1)

Let tangents at A and B intersect at C(h,k).
Then AB is the chord of contact for E2 w.r.t. point C.
So, equation of AB is T=0
i.e., hx16+ky48=1 (2)

Comparing (1) and (2), we get
cosθ2=h16, sinθ6=k48
cosθ=h8, sinθ=k8
h2+k2=64
p2+32=64 [(h,k)(p,3)]
p=55, (p>0)

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