Cambridge O Level Additional Mathematics · Syllabus 4037 · Coordinate Geometry of the Circle
Equation of a Circle
What is Equation of a Circle?
The relationship satisfied by the coordinates of every point on a circle and by no other point. A circle is the locus of points at a fixed distance, the radius, from a fixed point, the centre, so applying the distance formula and squaring both sides gives the standard form: a point with coordinates x and y lies on the circle of centre a comma b and radius r exactly when x minus a all squared plus y minus b all squared equals r squared. The number subtracted inside each bracket is the corresponding coordinate of the centre, so the centre coordinates appear with the opposite sign to the constants visible in the equation, and the quantity on the right-hand side is the square of the radius rather than the radius itself. This form is given in the List of formulas in the Cambridge Additional Mathematics 4037 examination papers.
This definition is part of the Coordinate Geometry of the Circle chapter in Cambridge O Level Additional Mathematics.
Equation of a Circle in context
A circle is the set of all points at a fixed distance from a fixed point. Every equation, every tangent and every intersection in this chapter is that one sentence written in algebra. The equation of a circle is the distance formula, squared so that no square root has to be carried around.
Questions students ask about Equation of a Circle
What is the standard equation of a circle?
\((x-a)^2+(y-b)^2=r^2\), where \((a,b)\) is the centre and \(r\) is the radius. It comes from squaring both sides of the distance formula \(\sqrt{(x-a)^2+(y-b)^2}=r\), which states that every point \((x,y)\) on the circle is exactly a distance \(r\) from the centre. Squaring removes the square root, which is why the right-hand side is \(r^2\) rather than \(r\) itself. This form is given in the List of formulas.