Let P(h,k) be the focus of parabola.
Circle C1 meets y−axis at (0,±2)
An arbitrary tangent to C2 is xcosθ+ysinθ=3
Parabola passes through (0,±2) and its directrix is
xcosθ+ysinθ−3=0
We know that in a parabola, distance between a point and the directrix and the distance between focus and the point are equal.
⇒h2+(k−2)2=(2sinθ−3)2 ⋯(1)
and h2+(k+2)2=(2sinθ+3)2 ⋯(2)
From equations (1) and (2),
k3=sinθ, h√5=cosθ
Since sin2θ+cos2θ=1
k2(3)2+h2(√5)2=1
∴ Required locus is x25+y29=1
Eccentricity, e=√1−59
⇒e=23
⇒12e=8