Comprehension- I A coplanar beam of light emerging from a point source have equation `lambda x-y+2(1+lambda)=0,lambda in R.` The rays of the beam strike an elliptical surface and get reflected. The reflected rays form another convergent beam having equation `mu x-y+2(1-mu)=0,mu in R.` Foot of the perpendicular from the point (2, 2) upon any tangent to the ellipse lies on the circle `x^2 + y^2 - 4y - 5 = 0` The eccentricity of the ellipse is equal to

Options

A. `1//3`
B. `1//sqrt(3)`
C. `2//3`
D. none of these

Explanation

`lambdax-y+2(1+lambda)=0`
or `lambda(x+2)-(y-2)=0,lambda in R`
This line passes through `(-2,2)`
`mux-y+2(1-mu)=0, mu in R`
or `mu(x-2)-(y-2)=0`
This line passes through (2,2)
Clearly, (-2,2) and (2,2) reprsente the foci of the ellipse So, 2ae=4
The circle `x^(2)+y^(2)-4y-5=0i.e.,x^(2)+(y-2)^(2)=9` respresent an auxiliar circle. Thus, `a^(2)=9 or e=2//3 and b^(2)=5`