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Question

The equation of the circle through the points of intersection of x2+y21=0,x2+y22x4y+1=0 and touching the line x + 2y = 0, is


A

x2+y24x4y3=0

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B

3(x2+y2)+4x4y3=0

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C

x2+y2x2y=0

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D

3(x2+y2)+4(x+y)3=0

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Solution

The correct option is C

x2+y2x2y=0


Family of circles is

x2+y22x4y+1+λ(x2+y21)=0.

(1+λ)x2+(1+λ)y22x4y+(1λ)=0

x2+y221+λx41+λy+1λ1+λ=0 ............(i)

Centre is [11+λ,21+λ]

and radius = (11+λ)2+(21+λ)21λ1+λ=4+λ21+λ.

Since it touches the line x + 2y = 0, hence

Radius = Perpendicular from centre to the line

i.e., 11+λ+221+λ12+22 = 4+λ21+λ

5=4+λ2=±1

λ=1 cannot be possible in case of circle. So λ=1.

Thus, from (i) x2+y2x2y=0 is the required equation of the circle.


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