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Comprehension- I A coplanar beam of ligh...

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

A

`1//3`

B

`1//sqrt(3)`

C

`2//3`

D

none of these

Text Solution

Verified by Experts

`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`
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