Question

Equation of the circle which touches the coordinate axes and passes through the point $(2,1)$ is ?

• A. $x^2+y^2+8x+8y-29=0$
• B. $x^2+y^2-10x-10y+25=0$
• C. $x^2+y^2+10x+10y-35=0$
• D. $x^2+y^2-20x-20y+55=0$

Answer 1

The correct option is B $x^2+y^2-10x-10y+25=0$

Let the required equation be $x^2+y^2+2gx+2fy+c=0$

It passes through the point $(2,1)$

$⟹4+1+4g+2f+c=04g+2f+c+5=0\cdots \left(1\right)$

It touches $\text{x-axis}$

$⟹g^2-c=0g^2=c\cdots \left(2\right)$

It touches $\text{y-axis}$

$⟹f^2-c=0f^2=c\cdots \left(3\right)$

From $\left(2\right),\left(3\right)$

$g^2=f^2⟹g=f\cdots \left(4\right)$

Put $\left(2\right)\left(4\right)$ in $\left(1\right)$

$⟹4g+2g+g^2+5=0g^2+6g+5=0(g+1)(g+5)=0g=-1,-5$

Put $g=-5⟹f=-5⟹c=25$

So the required Equation is $x^2+y^2-10x-10y+25=0$