How do you find the equation of a tangent to a circle at a given point?

The geometry behind circle tangents

A tangent line touches a circle at exactly one point, known as the point of contact. A fundamental theorem in Euclidean geometry states that the radius drawn to the point of contact is perpendicular to the tangent line at that point.

Because perpendicular lines in coordinate geometry have slopes that are negative reciprocals, knowing the coordinates of the center and the contact point gives you everything needed to determine the equation of a tangent to a circle at a given point.

Step-by-step method to find the tangent equation

First, determine the coordinates of the center (h, k) from the circle equation (x - h)^2 + (y - k)^2 = r^2. If the circle is in general form, complete the square to identify the center.

Next, calculate the slope of the radius, m_radius, using the slope formula (y1 - k) / (x1 - h), where (x1, y1) is the contact point. The slope of the tangent line is the negative reciprocal: m_tangent = -1 / m_radius.

Finally, substitute m_tangent and the contact point (x1, y1) into the point-slope form: y - y1 = m_tangent * (x - x1). Simplify into slope-intercept form (y = mx + b) or standard form (Ax + By + C = 0).

A worked example and special cases

Consider the circle x^2 + y^2 = 25 and the point (3, 4) on it. The center is (0, 0). The slope of the radius is (4 - 0) / (3 - 0) = 4/3. The slope of the tangent line is the negative reciprocal, -3/4. Using point-slope form gives y - 4 = -3/4 * (x - 3), which rearranges to 3x + 4y - 25 = 0.

Watch for vertical and horizontal lines. If the radius is vertical (m is undefined), the tangent is horizontal with equation y = y1. If the radius is horizontal (m = 0), the tangent is vertical with equation x = x1.