Question

The equation of the circle passing through $\left(4,5\right)$ and having center $\left(2,2\right)$ is

• A. $x^2+y^2+4x+4y-5=0$
• B. $x^2+y^2-4x-4y-5=0$
• C. $x^2+y^2-4x=13$
• D. $x^2+y^2-4x-4y+5=0$

Answer 1

The correct option is B

$x^2+y^2-4x-4y-5=0$

Explanation for the correct option:

Finding the equation of the circle:

Given that the circle passes through a point $\left(4,5\right)$ having center $\left(2,2\right)$,

The general equation of the circle is $x^2+y^2+2gx+2fy+c=0$ where the centre is $\left(-g,-f\right)$.

By comparing with the given center we get,

$g=f=-2$

Substituting the values we get the equation of the circle as

$x^2+y^2-4x-4y+c=0$

Since the circle passes through $\left(4,5\right)$, substituting the values of $\left(x,y\right)$ as $\left(4,5\right)$ in the above equation we get

$$\begin{gathered} 16+25-16-20+c=0 \\ \Rightarrow c=-5 \end{gathered}$$

Therefore, the equation of the circle will be

$x^2+y^2-4x-4y-5=0$

Hence, option (B) is the correct answer.