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A variable tangent to the circle x^(2)+y...

A variable tangent to the circle `x^(2)+y^(2)=1` intersects the ellipse `(x^(2))/(4)+(y^(2))/(2)=1` at point P and Q. The lous of the point of the intersection of tangents to the ellipse at P and Q is another ellipse. Then find its eccentricity.

Text Solution

Verified by Experts

The correct Answer is:
`sqrt(3)//2`
Equation of tangent to the circle `x^(2)/(4)+y(2)=1` is `xcos theta+y sin theta=1" "(1)`
This line meets the ellipse `(x^(2))/(4)+(y^(2))/(2)=1` at P and Q.
Let the tangents to ellipse at P and Q intersect at R(h,k) PQ is chood of contact of ellipse w.r.t, point R.
So, its equations is
`hx+2ky=4" "(2)`
Equations (1) and (2) represent hte same straight line.
`:. h=cos, theta, k=2 sin theta`
Eliminating `theta,` we get locus as `(x^(2))/(16)+(y^(2))/(2=4)=1` which is ellipse.
`e=sqrt((16-4)/(16))=(sqrt(3))/(2)`
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