Circle with centre \((a, b)\) and radius \(r\):
\((x-a)^2 + (y-b)^2 = r^2\)
Finding centre and radius: Rearrange (often by completing the square) into this form. The numbers in brackets have opposite signs to the centre coordinates. Remember to take the square root of the right-hand side for \(r\).
General form: \(x^2 + y^2 + 2fx + 2gy + c = 0\)
Complete the square for \(x\) and \(y\) to get \((x+a)^2 + (y+b)^2 = r^2\) and identify centre \((-a, -b)\) and radius \(r\).
For a right-angled triangle, the hypotenuse is a diameter of its circumcircle. Therefore:
Any angle at the circumference in a semicircle is a right angle.
A tangent touches the circle at exactly one point. The tangent is perpendicular to the radius at the point of contact.
To find tangent at P: (1) Gradient of radius OP; (2) Tangent gradient = negative reciprocal; (3) Equation of line through P with that gradient.
