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Q. An ellipse has eccentricity 12 and focus at the point P(12,1) its one directrix is the common tangent at the point P, to the circle x2+y2=1 and the hyperbola x2y2=1. The equation of the ellipse in standard form is

Conic Sections

Solution:

image
Common tangent to the circle x2+y2=1
and Hyperbola x2y2=1
is x=1
Point P(12,1), equation of the directrix x=1
Ellipse : PS=epm
(x12)2+(y1)2=12(x1)
(x12)2+(y1)2=14(x1)2
After simplification
9(x13)2+12(y1)2=1