Question

Find the standard equation of the circle whose center is the point $\left(-1,2\right)$if $\left(3,5\right)$is a point on the circle.

Short answer

The equation of circle is$x^2+y^2+2x-4y-1=0$

Step-by-step solution

Step 1. Given Information

Centre of circle $\left(h,k\right)=\left(-1,2\right)$and $\left(3,5\right)$is on the circle.

Step 2. Finding radius

The given points are $\left(-1,2\right)$and $\left(3,5\right)$

$$\begin{gathered} r=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2} \\ r=\sqrt{{\left(3+1\right)}^2+{\left(5-2\right)}^2} \\ r=\sqrt{16+9}=\sqrt{25}=5 \end{gathered}$$

Step 3. Finding Equation of Circle

The given centre is $\left(h,k\right)=\left(-1,2\right)$

$$\begin{gathered} {\left(x+1\right)}^2+{\left(y-2\right)}^2=5 \\ {x}^2+2x+1+y^2-4y+4=5 \\ {x}^2+y^2+2x-4y-1=0 \end{gathered}$$