The equation of a circle

A circle is all the points at a given distance – the radius $\displaystyle R$ – from a given point, its centre. When the centre is the origin $\displaystyle O(0,0)$, the distance from $\displaystyle O$ to a point $\displaystyle (x,y)$ is $\displaystyle \sqrt{x^{2}+y^{2}}$, so the points of the circle are those with $\displaystyle {\sqrt{x^{2}+y^{2}}}={R}$, or, squared:

• This is the canonical circle – its centre at the origin. Here $\displaystyle {R}={5}$: $\displaystyle {3^{2}+4^{2}}={25}$, so $\displaystyle A(3,4)$ is on it.
• The circle is not the graph of a function: $\displaystyle {x}={3}$ has two points on it, $\displaystyle A(3,4)$ and $\displaystyle B(3,-4)$.

The circle with centre $\displaystyle M(a,b)$ and radius $\displaystyle R$ – the points whose distance from $\displaystyle M$ is $\displaystyle R$:

• It is the canonical circle shifted $\displaystyle a$ to the right and $\displaystyle b$ up, as $\displaystyle {y}={(x-3)^{2}}$ is the parabola $\displaystyle {y}={x^{2}}$ shifted $\displaystyle 3$ to the right.
• In the drawing: the dashed circle $\displaystyle {x^{2}+y^{2}}={9}$, and the same circle shifted to the centre $\displaystyle (3,2)$.
• Reading the centre: in $\displaystyle {(x+1)^{2}+(y-4)^{2}}={7}$, $\displaystyle {x+1}={x-(-1)}$, so the centre is $\displaystyle (-1,4)$ and the radius $\displaystyle \sqrt{7}$.

Write the centre and the radius of each circle.

1. $\displaystyle {(x-3)^{2}+(y+2)^{2}}={16}$
2. $\displaystyle {x^{2}+(y-1)^{2}}={5}$
3. $\displaystyle {(x+4)^{2}+y^{2}}={\frac{9}{4}}$

Write the equation of the circle with centre $\displaystyle (2,-1)$ and radius $\displaystyle 3$.

Substitute the point in the left side of the equation and compare with $\displaystyle R^{2}$: the left side is the square of the point’s distance from the centre.

• Equal to $\displaystyle R^{2}$ – the point is on the circle;
• less – inside it;
• more – outside it.

Example: the circle $\displaystyle {(x-1)^{2}+(y+2)^{2}}={10}$.

• $\displaystyle (4,-1)$: $\displaystyle {3^{2}+1^{2}}={10}$ – on the circle;
• $\displaystyle (2,0)$: $\displaystyle {1^{2}+2^{2}}={5<10}$ – inside it;
• $\displaystyle (5,1)$: $\displaystyle {4^{2}+3^{2}}={25>10}$ – outside it.

The circle: $\displaystyle {(x-2)^{2}+(y+3)^{2}}={25}$.

Where is each of the points: on the circle, inside it or outside it?

$\displaystyle AB$ is a diameter of a circle: $\displaystyle A(-3,1)$ and $\displaystyle B(5,7)$.
Write the equation of the circle.
(A hint: the centre is the midpoint of the diameter.)

A square is inscribed in a canonical circle – its four vertices are on the circle.
One of the vertices of the square is $\displaystyle (3,4)$.

Write the equation of the circle.

Find the other three vertices of the square.
(A hint: the diagonals of a square are equal, perpendicular, and bisect each other – so they are diameters.)

The centre of a circle is on the $\displaystyle x$-axis, and the circle passes through $\displaystyle A(1,3)$ and $\displaystyle B(5,1)$.

Write the equation of the circle.