Circle equation calculator

Find general equation of a circle with a center at $ \color{blue}{C = \left( -3 , 9 \right)} $ and radius $ \color{blue}{ 7 }$

solution

The circle equation is:

$$ \color{blue}{ x^2 + y^2 + 6 x - 18 y +41 = 0 } $$

explanation

Step 1: The equation of a circle with center at $(\color{blue}{h} , \color{red}{k})$ and a radius of $ r $ is:

$$ (x - \color{blue}{h} )^2 + (y - \color{red}{k})^2 = r^2 $$

In this example $ \color{blue}{h = -3 } $, $ \color{red}{k = 9 }$ and $ r = 7 $, so after substituting into above formula we have:

$$ \begin{aligned} \left(x - \left( -3 \right) \right)^2 + \left(y - 9 \right)^2 &= 7^2 \\ \left( x + 3 \right)^2 + \left( y - 9 \right)^2 &= 49 \end{aligned} $$

Step 2: Now we will find general form.

To find general form we will expand the standard form and bring all terms to the left side.

$$ \begin{aligned} \left( x + 3 \right)^2 + \left( y - 9 \right)^2 &= 49\\ x^2 + 6 x + 9 + y^2 - 18 y + 81 -49 &= 0 \\ x^2 + 6 x + y^2 - 18 y +41 &= 0 \end{aligned} $$