Cambridge O Level Additional Mathematics · Syllabus 4037 · Coordinate Geometry of the Circle
Tangent to a Circle
What is Tangent to a Circle?
A straight line that meets a circle at exactly one point, called the point of contact. Algebraically, substituting a tangent into the circle equation produces a quadratic with a repeated root, so its discriminant is zero, and the perpendicular distance from the centre of the circle to a tangent is exactly equal to the radius. Geometrically, the radius drawn to the point of contact is perpendicular to the tangent there, which means the tangent gradient is the negative reciprocal of the radius gradient whenever both are defined. That perpendicularity is the standard method for finding a tangent equation in Cambridge Additional Mathematics 4037, where no use of calculus is expected for this outcome. From a point outside the circle exactly two tangents can be drawn, and they have equal lengths from that point to their respective points of contact.
This definition is part of the Coordinate Geometry of the Circle chapter in Cambridge O Level Additional Mathematics.
Questions students ask about Tangent to a Circle
How do you find the equation of a tangent to a circle at a given point?
First check the point actually lies on the circle. Find the gradient of the radius from the centre to that point, then take its negative reciprocal, since the radius and the tangent are always perpendicular at the point of contact. Use \(y-y_1=m(x-x_1)\) with the point and the new gradient, and simplify. At the top or bottom of a circle the tangent is horizontal; at the left or right extremes it is vertical and has no gradient.