Circles — Formula Sheet

(x - h)^2 + (y - k)^2 = r^2

x^2 + y^2 = r^2

x^2 + y^2 + 2gx + 2fy + c = 0

(x - x₁)(x - x₂) + (y - y₁)(y - y₂) = 0

x = h + r cosθ, y = k + r sinθ

x = r cosθ, y = r sinθ

The standard form of the equation of a circle with centre (h, k) and radius r is: (x - h)² + (y - k)² = r².

The general form of the equation of a circle is

x² + y² + 2gx + 2fy + c = 0.

From the general form, the centre is (-g, -f) and the radius is √(g² + f² - c).

When the endpoints of a diameter are (x₁, y₁) and (x₂, y₂), the equation of the circle is (x - x₁)(x - x₂) + (y - y₁)(y - y₂) = 0.

The equation of the tangent to the circle S = 0 at point (x₁, y₁) is S₁ = 0.

Two circles are orthogonal if the angle between them is 90°, which happens when 2(gg' + ff') = c + c'.

Standard form

$(x - h)^2 + (y - k)^2 = r^2$

General form

$x^2 + y^2 + 2gx + 2fy + c = 0$