Question

The centre and radius of the circle passing through the points $(1,0),(3,-2)$ and $(-1,-2)$ are given by

• A. $(1,-2);2$
• B. $(1,2);2$
• C. $(-1,-2);1$
• D. $(-1,2);1$

Answer 1

The correct option is A $(1,-2);2$

The circle passes through $A(1,0),B(3,-2)$ and $C(-1,-2)$

Let the general equation of circle be $x^2+y^2+2gx+2fy+c=0$ ...... $\left(i\right)$

Since, the circle passes through $A,B$ and $C$, all three points satisfy $\left(i\right)$

At $A(1,0)$

$⟹1^2+0^2+2g\left(1\right)+2f\left(0\right)+c=0$

$⟹2g+c=-1$ ...... $\left(ii\right)$

At $B(3,-2)$

$⟹3^2+(-2{)}^2+2g(3)+2f(-2)+c=0$

$⟹6g-4f+c=-13$ ...... $\left(iii\right)$

At $C(-1,-2)$

$⟹(-1{)}^2+(-2{)}^2+2g(-1)+2f(-2)+c=0$

$⟹2g+4f-c=5$ ...... $\left(iv\right)$

Solving $\left(iii\right)$ and $\left(iv\right)$ simultaneously, we get

$g=-1$

Substituting value of $g$ in $\left(ii\right)$ we get

$c=1$

And then substituting values of $g,c$ in $\left(iii\right)$ we get

$f=2$

Now, put all the values i.e. $f,g$ and $c$ in $\left(i\right)$, we get

$x^2+y^2+2(-1)x+2\left(2\right)y+1=0$

$⟹x^2+y^2-2x+4y+1=0$

$⟹(x-1{)}^2+(y+2{)}^2=4$ is the required equation of circle with centre $(-g,-f)$ and radius $\sqrt{g^2+f^2-c}$

$\therefore$ Centre of the circle is $(1,-2)$ and radius is $2$