Question

Let $O$ be the origin and $P$ be a variable point on the circle $x^2+y^2+2x+2y=0$. If the locus of mid-point of $OP$ is $x^2+y^2+2gx+2fy=0$, then the value of $(g+f)$ is equal to

• A. $-1$
• B. $1$
• C. $2$
• D. $-2$

Answer 1

The correct option is B $1$

Let the point $P$ be $(\alpha ,\beta )$, and it is a variable point on circle $S_1:x^2+y^2+2x+2y=0$, then point $P$ also satisfies the circle,
$\therefore {\alpha }^2+{\beta }^2+2\alpha +2\beta =0$........{1}
Mid-point of origin $O(0,0)$ and $P(\alpha ,\beta )$ is $M(\frac{\alpha }{2},\frac{\beta }{2})$,
The locus of $M$ is $x^2+y^2+2gx+2fy=0$, which means $M$ must satisfy this.
$\therefore {\alpha }^2+{\beta }^2+4g\alpha +4f\beta =0$......{2}
On comparing the coefficients of {1} and {2}, we get
$g=\frac{1}{2},f=\frac{1}{2}$
$\therefore g+f=1$