Question

The points of intersection of two ellipses \(x^{2}+2 y^{2}-6 x-12 y+20=0\) and \(2 x^{2}+y^{2}-10 x-6 y+15=0\) lie on a circle. The centre of the circle is
(A) \((8,3)\)
(B) \((8,1)\)
(C) \(\left(\frac{8}{3}, 3\right)\)
(D) \((3,8)\)

Answer 1

Ans: (C)
Hint \(: x^{2}+2 y^{2}-6 x-12 y+20+\lambda\left(2 x^{2}+y^{2}-10 x-6 y+15\right)=0\)

\(\Rightarrow(1+2 \lambda) x^{2}+(2+\lambda) y^{2}-(10 \lambda+6) x-(6 \lambda+12) y+15 \lambda+20=0\)
\(2+\lambda=1+2 \lambda\)
\(\Rightarrow \lambda=1\)
\(\therefore\) Centre : \(\left(\frac{8}{3}, 3\right)\)

Answer 2

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Answer 3

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