The tangent line to a circle

A tangent to a circle is a line that touches the circle at exactly one point. Here we find the tangent to the circle $x^2 + y^2 = 25$ (center $O(0, 0)$, radius $5$) at the point $(4, 3)$ on it. First we check that $(4, 3)$ is on the circle: $4^2 + 3^2 = 16 + 9 = 25$, so it is.

What sets the direction of the tangent is the radius to the point of tangency. A tangent meets that radius at a right angle. The radius $O(0, 0) \to (4, 3)$ has slope $\dfrac{3}{4}$, so the tangent has the perpendicular slope $-\dfrac{4}{3}$.

The line through $(4, 3)$ with slope $-\dfrac{4}{3}$ is $y - 3 = -\dfrac{4}{3}(x - 4)$, which rearranges to $4x + 3y = 25$.

In general, the tangent to the circle $x^2 + y^2 = r^2$ (centered at the origin, radius $r$) at a point $(x_1, y_1)$ on it can be written as follows.

Putting $(x_1, y_1) = (4, 3)$ and $r^2 = 25$ gives $4x + 3y = 25$, matching the result above. The large dots on the graph are the center $O(0, 0)$ and the point of tangency $(4, 3)$; you can see the radius and the tangent meeting at a right angle.