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Circles in Coordinate Geometry

Lesson · Beginner

Geometry: Coordinate Geometry

Equation of a Circle

Circle with center and radius

A circle with center (x₀, y₀) and radius r:

(x−x0)2+(y−y0)2=r2(x-x_0)^2+(y-y_0)^2=r^2

Finding the Equation of a Tangent Line

Tangent line to a circle

Approach 1

Let the tangent line through the point P(x₁, y₁) have slope m.

ℓ: y−y1=m(x−x1)\ell:\ y-y_1=m(x-x_1)
(x−x0)2+(y−y0)2=r2(x-x_0)^2+(y-y_0)^2=r^2

Substitute the line equation into the circle equation. Since a tangent line intersects the circle at exactly one point, set the discriminant equal to zero and solve for m.

Δ=0\Delta=0

Approach 2

Suppose the circle has center (x₀, y₀) and radius r, and the line is

Ax+By+C=0Ax+By+C=0

The distance from the center to the line is

∣Ax0+By0+C∣A2+B2\frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}

The line is tangent to the circle exactly when

∣Ax0+By0+C∣A2+B2=r\frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}=r

This is often faster than substituting and setting the discriminant equal to zero.

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