Circle

A circle is the set of points equidistant from a fixed point (the centre). The fixed distance is the radius. It is the simplest conic , eccentricity $0$ , and the building block for many later results.

Standard form

If the centre is the origin and the radius is $r > 0$, then a point $(x, y)$ on the circle satisfies $\boxed{x^2 + y^2 = r^2.}$

If the centre is $(h, k)$, shift coordinates: $\boxed{(x - h)^2 + (y - k)^2 = r^2.}$

General form

Expanding $(x - h)^2 + (y - k)^2 = r^2$: $x^2 + y^2 - 2 h x - 2 k y + (h^2 + k^2 - r^2) = 0.$

In general: $\boxed{x^2 + y^2 + 2 g x + 2 f y + c = 0,}$ with centre $(-g, -f)$ and radius $\sqrt{g^2 + f^2 - c}$ (provided $g^2 + f^2 - c > 0$).

If $g^2 + f^2 - c = 0$, the "circle" degenerates to a single point $(-g, -f)$. If $g^2 + f^2 - c < 0$, no real circle.

Finding a circle

Three points determine a circle (provided they are not collinear). Substitute each point into the general form; you get three linear equations in $g, f, c$. Solve.

Equation of a circle in diameter form. If $A(x_1, y_1)$ and $B(x_2, y_2)$ are endpoints of a diameter: $(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0.$

(Reason: angle in a semicircle is a right angle.)

Position of a point relative to a circle

For a circle $x^2 + y^2 + 2 g x + 2 f y + c = 0$ and a point $P(x_0, y_0)$, compute $S = x_0^2 + y_0^2 + 2 g x_0 + 2 f y_0 + c.$

• $S > 0$: $P$ is outside.
• $S = 0$: $P$ is on the circle.
• $S < 0$: $P$ is inside.

Worked examples

Example 1. Find the centre and radius of $x^2 + y^2 - 6 x + 8 y - 11 = 0$.

Compare: $2 g = -6 \Rightarrow g = -3$; $2 f = 8 \Rightarrow f = 4$; $c = -11$. Centre: $(-g, -f) = (3, -4)$. Radius: $\sqrt{9 + 16 + 11} = \sqrt{36} = 6$.

Example 2. Equation of the circle with centre $(2, -3)$ and radius $5$.

$(x - 2)^2 + (y + 3)^2 = 25 \Rightarrow x^2 + y^2 - 4 x + 6 y - 12 = 0$.

Example 3. Find the circle passing through $(0, 0), (3, 0), (0, 4)$.

General form $x^2 + y^2 + 2 g x + 2 f y + c = 0$. At $(0,0)$: $c = 0$. At $(3, 0)$: $9 + 6 g = 0 \Rightarrow g = -3/2$. At $(0, 4)$: $16 + 8 f = 0 \Rightarrow f = -2$.

Equation: $x^2 + y^2 - 3 x - 4 y = 0$. Centre $(3/2, 2)$, radius $\sqrt{9/4 + 4} = 5/2$.

Example 4. Equation of the circle with diameter endpoints $(1, 2)$ and $(5, 6)$.

$(x - 1)(x - 5) + (y - 2)(y - 6) = 0 \Rightarrow x^2 - 6 x + 5 + y^2 - 8 y + 12 = 0 \Rightarrow x^2 + y^2 - 6 x - 8 y + 17 = 0$.

Example 5. Is the point $(2, 3)$ inside the circle $x^2 + y^2 = 16$?

$S = 4 + 9 - 16 = -3 < 0$. Yes, inside.

Try it yourself

1. Find the centre and radius of $(x + 1)^2 + (y - 2)^2 = 9$.
2. Find the centre and radius of $x^2 + y^2 - 4 x - 6 y - 12 = 0$.
3. Equation of circle with centre $(-3, 5)$ and radius $\sqrt{7}$.
4. Find the circle through $(0, 0), (4, 0), (0, 6)$.
5. Equation of circle with diameter from $(-1, 2)$ to $(3, -4)$.
6. Is $(1, 1)$ inside, on, or outside $x^2 + y^2 - 2 x - 2 y - 7 = 0$?
7. Find the circle of radius $5$ touching the $x$-axis at $(3, 0)$.
8. Find the circle passing through $(1, 0), (-1, 0), (0, 1)$.
9. If a circle's centre is on $y = x$ and passes through $(2, 3)$ and $(0, 1)$, find its equation.
10. Find the equation of the circle with centre $(0, 0)$ that passes through $(3, 4)$.
11. A circle has equation $x^2 + y^2 - 8 x + 6 y + c = 0$. For what range of $c$ does it represent a real circle?
12. Find the largest circle contained inside the rectangle with vertices $(\pm 5, \pm 3)$.

Pitfalls / Tricks

• Coefficients of $x^2$ and $y^2$ must be equal (and same sign) for an equation to represent a circle.
• There is no $xy$ term in a circle (otherwise it is rotated , not standard).
• For an equation to be a real circle, $g^2 + f^2 - c > 0$.
• Insight. A circle is a special ellipse with $a = b$. Many problems on ellipses degenerate cleanly when $a = b$.