Question

Equation of the circle with centre on the $y$-axis and passing through the origin and the point $(2,3)$ is

• A. $x^2+y^2+13y=0$
• B. $3x^2+3y^2+13x+3=0$
• C. $6x^2+6y^2-26y=0$
• D. $x^2+y^2+13x+3=0$

Answer 1

Answer: (C)

Let the centre be $\left(0,k\right)$.
The equation of circle be
$x^2+y^2-2\left(0\right)x-2ky+c=0$
$\Rightarrow x^2+y^2-2ky+c=0\dots \left(i\right)$
Since, it passes through $\left(0,0\right)$ and $\left(2,3\right)$.
$\therefore 0^2+0^2-2k\left(0\right)+c=0$
$\Rightarrow c=0$
and ${\left(2\right)}^2+{\left(3\right)}^2-2k\left(3\right)+c=0$
$\Rightarrow 13-6k+c=0$
$\Rightarrow 13-6k+0-0=0$
$\Rightarrow k=\frac{13}{6}$
On putting the values of $k$ and $c$ in $\left(i\right)$, we get
$x^2+y^2-2\left(\frac{13}{6}\right)y+0=0$
$\Rightarrow 6x^2+6y^2-26y=0$