Question

The equation of the circle passing through three points $(1,0),$ $(-1,0)$ and $(0,1),$ is

• A. $x^2+y^2+2x+2y=1$
• B. $x^2+y^2+2x=1$
• C. $x^2+y^2+2y=1$
• D. $x^2+y^2=1$

Answer 1

The correct option is D $x^2+y^2=1$

Let the equation of the general form of the required circle be $x^2+y^2+2gx+2fy+c=0\dots \left(1\right)$
According to the problem, the above equation of the circle passes through the points $(1,0),(-1,0)$ and $(0,1).$ Therefore,
$1+2g+c=0\dots \left(2\right)$
$1-2g+c=0\dots \left(3\right)$
$1+2f+c=0\dots \left(4\right)$
Solving equations $\left(2\right),\left(3\right)$ and $\left(4\right)$, we get
$g=0,f=0,c=-1$

Substituting these values in equation $\left(1\right),$ we obtain the equation of the required circle as $x^2+y^2=1.$