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Question

# From any point on the line y=x+4, tangents are drawn to the auxiliary circle of the ellipse x2+4y2=4. If P,Q are the points of contact and A,B are the corresponding points of P and Q on the ellipse respectively, then the locus of the midpoint of AB is

A
4x2+y2+y2x=0
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B
4x2+y2+y+2x=0
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C
4y2+x2+x2y=0
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D
4y2+x2+x+2y=0
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Solution

## The correct option is C 4y2+x2+x−2y=0Given equation of ellipse is x24+y21=1 …(1) auxiliary circle is x2+y2=4 …(2) Let P≡(x1,y1), Q≡(x2,y2), A≡(x′1,y′1), B≡(x′2,y′2) Now, x1=x′1 and x2=x′2 By equation (1) and (2) and y1=2y′1, y2=2y′2 and M(h,k) be the midpoint of AB. any point on the line y=x+4 will be R(t,t+4) Equation of chord PQ is tx+(t+4)y=4⇒x=4−(t+4)yt from equation (2) 16+(t+4)2y2−8(t+4)y+t2y2=4t2⇒[t2+(t+4)2]y2−8(t+4)y+16−4t2=0 y1+y2=8(t+4)t2+(t+4)2 similarly by replacing y we get x1+x2=8tt2+(t+4)2 Now h=x1+x22=4tt2+(t+4)2 k=y′1+y′22=y1+y24=2(t+4)t2+(t+4)2 ∴h24+k2=4[t2+(t+4)2][t2+(t+4)2]2=4t2+(t+4)2 and k2−h4=4t2+(t+4)2 Hence locus will be x24+y2=y2−x4 4y2+x2+x−2y=0.

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