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Q. If the tangent at the point (4cosθ,1611sinθ) to the ellipse 16x2+11y2=256 is also a tangent to the circle x2+y22x=15, then the value of θ is

Conic Sections

Solution:

The equation of the tangent at (4cosθ,1611sinθ) to
the ellipse 16x2+11y2=256 is
16x(4cosθ)+11y(1611sinθ)=256
or 4xcosθ+11ysinθ=16
This touches the circle (x1)2+y2=42 if
|4cosθ1616cos2θ+11sin2θ|=4
(cosθ4)2=16cos2θ+11sin2θ
15cos2θ+11sin2θ+8cosθ16=0
4cos2θ+8cosθ5=0
(2cosθ1)(2cosθ+5)=0
cosθ=12
θ=±π3