The Equation of a Tangent at a Point on a Circle

Understanding the Tangent Line

A tangent to a circle is a straight line that just touches the circle at one point.
The tangent is perpendicular to the radius at the point of contact.
Generally, the equation of a tangent will be in the form of y = mx + c.

The gradient of the radius can be determined by considering the coordinates of the center of the circle and the point of tangency. Since the radius is perpendicular to the tangent, knowing the gradient of the radius allows us to find the gradient of the tangent line, as the gradients of perpendicular lines multiply together to give -1.
The equation of the tangent is given by the formula (x - x₁)x + (y - y₁)y = r², where (x₁, y₁) are the coordinates of the point where the line touches the circle and r is the radius of the circle.

For example, if the circle’s equation is (x-2)² + (y+1)² = 9 and the point of tangency is (5, -4), you can substitute these values into the equation to find the equation of the tangent line.
In problems that involve finding the equation of a tangent to a circle, always start by finding out as much as you can about the circle and the point where the line touches the circle. Then use the tangent equation to find the line’s equation.

The tangent to a circle is always perpendicular to the radius of the circle at the point of tangency.
The radius and tangent meet at a 90 degree angle.
The tangent only touches the circle at one point.