Question

If the equation of circle touching the $Y-$ axis at $(0,3)$ and making an intercept of $8$ unit on Xaxis is $x^2+y^2+2gx+2fy+c=0$, then $(g+f+c)$ can be

• A. $1$
• B. $7$
• C. $11$
• D. $14$

Answer 1

The correct option is A $1$

Equation of circle touches y-axis at (0,3) is

$\begin{array}{c}{\left(x-a\right)}^2+{\left(y-3\right)}^2=a^2\\ x^2+y^2-2ax-6y+9=0\end{array}$

Since the length of intercept on x-axis at this circle is 8 unit.

$\begin{array}{c}2\sqrt{\left(a^2-9\right)}=8\\ \sqrt{a^2-9}=4\\ a^2-9=16\\ a^2=16+9\Rightarrow a=\sqrt{25}=5\left(a>0\right)\end{array}$

required equation is

$x^2+y^2-10x-6y+9=0$

compare this with given equation, given

$\begin{array}{c}g=-5;f=-5c=9\\ \therefore g+f+=-5-3+9=1\end{array}$