We have seen that the gradient and the y-intercept of a straight line are what make it special. In order to represent circles, we need to acknowledge that the following make one circle different from another:
- the center (two circles can have the same size but a different offset)
- the radius (concentric circles have the same center but different radius)
The standard form
Given that a circle has a center $(a,b)$ and radius $r$ units, the standard form of the equation of the circle is: $$ \begin{equation}\begin{aligned} (x-a)^2+(y-b)^2=r^2\\ \end{aligned}\end{equation} $$
Challenge
Determine the center and radius of the following circles:
- $(x-1)^2+(y-3)^2=25$
- $(x+10)^2+(y-2)^2=36$
- $(x-7)^2+(y-4)^2=100$
- $(x-1)^2+(y+4)^2=\pi^2$
- $(x-9)^2+(y-1)^2=14$
The general form
When we expand the standard form and transpose all values to the one side of the equal sign, we get the general form: $$ \begin{equation}\begin{aligned} x^2+2fx+y^2+2gy+c=0\\ \end{aligned}\end{equation} $$
We can convert this to the standard equation by completing the squares or by rewriting from the formula: $$ \begin{equation}\begin{aligned} r^2=f^2+g^2-c\\ \end{aligned}\end{equation} $$
Where we get $f$ and $g$ from respectively dividing the coefficients of $x$ and $y$ by two ($2$). The center will be $(-f,-g)$.