Question

The equation of a circle which is co-axis with circles $2 x^{2}+2 y^{2}-2 x+6 y-3=0 \& x^{2}+y^{2}+4 x+2 y+1=0$ It is given that the center of the circle to be determined lies on the radical axis of these two circles, is.

• A. $4 x^{2}+4 y^{2}+6 x-10 y-1=0$
• B. $4 x^{2}+4 y^{2}+6 x+10 y+1=0$
• C. $4 x^{2}+4 y^{2}+6 x+10 y-1=0$
• D. None of these

Answer 1

Answer: (C)

Radical axis : $S _{1}- S _{2}=0$
and $S_{1}+\lambda\left(S_{2}-S_{1}\right)=0$
$10 x-2 y+5=0$ ...(i)
and $\left(x^{2}+y^{2}+4 x+2 y+1\right)+\lambda(10 x-2 y+5)=0$
From (i),
$-20-50 \lambda+2-2 \lambda+5=0$
$\Rightarrow \lambda=\frac{-1}{4}$
Required Circle:
$\left(x^{2}+y^{2}+4 x+2 y+1\right)-\frac{1}{4}(10 x-2 y+5)=0$
$\Rightarrow 4 x^{2}+4 y^{2}+16 x+8 y+4-10 x+2 y-5=0 $
$\Rightarrow 4 x^{2}+4 y^{2}+6 x+10 y-1=0$