The tangent line to a circle

A tangent to a circle is a line that touches it at exactly one point¹. Here we find the tangent to $x^2 + y^2 = 25$ , with centre $O(0, 0)$ and radius $5$ , at the point $(4, 3)$ on it. First we check that the point is on the circle: $16 + 9 = 25$ .

The radius sets the direction

What fixes the direction of the tangent is the radius to the point of tangency, which the tangent meets at a right angle.

Line	Slope
Radius $O \to (4, 3)$	$\dfrac{3}{4}$
Tangent	$-\dfrac{4}{3}$

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

A formula for the tangent

For the circle $x^2 + y^2 = r^2$ and a point $(x_1, y_1)$ on it, the tangent can be written directly.

x_1 x + y_1 y = r^2

Putting $(x_1, y_1) = (4, 3)$ and $r^2 = 25$ gives the same line. For a circle centred at $(a, b)$ the tangent is $(x_1 - a)(x - a) + (y_1 - b)(y - b) = r^2$ .

Three ways to see the tangency

View	What it gives
Geometry	the tangent is perpendicular to the radius
Algebra	a repeated root, $(x - 4)^2 = 0$
Distance	$\dfrac{\|-25\|}{\sqrt{4^2 + 3^2}} = 5$ , equal to the radius

Substituting the tangent into the circle and simplifying gives $x^2 - 8x + 16 = 0$ , whose only solution is $x = 4$ . A discriminant of $0$ is what tangency really means. Were the distance from the centre less than the radius the line would cut the circle twice; were it greater they would not meet.

From a point outside the circle, incidentally, there are always two tangent lines.

The large dots on the graph are the centre and the point of tangency; you can see the radius and the tangent meeting at a right angle.

Tangent lines to circles, Wikipedia