Question

The equation of the circle which touches all the sides of the square with sides $y=3,x=6,y=7$ and $x=10$ is $x^2+y^2+2gx+2fy+c=0$. Find the value of $g+f+c$

Answer 1

We will first construct the square and identify/draw the circle which touches its sides

If we can find the centre and radius of the circle, we can write its equation.

Let our square be ABCD and the centre of the square be O. We can easily see that O is also the circle.

$x=\frac{6+10}{2}=8andy-coordinate=\frac{3+7}{2}=5$

The x-coordinate of O is given by

So centre O $\equiv$ (8, 5)

Radius will be half of any side.

$\Rightarrow r=\frac{7-3}{2}=\frac{10-6}{2}=2$

$\Rightarrow$equation of circle

$\equiv (x-8{)}^2+(y-5{)}^2=2^2$

$\equiv x^2+y^2-16x-10y+64+25=4$

$\Rightarrow x^2+y^2-16x-10y+85=0$

This is given as $x^2+y^2+2gx+2fy+c=0$

$\Rightarrow g+f+c=-16-10+85$

$=59$