Equations of circles

Japanese school year: Math II

What you learn

This section focuses on deriving the equation of a circle from its center and radius, as well as finding tangents and intersection points with lines. It is widely used to analyze circular curves and geometric tangency algebraically. Familiarity with the distance formula and linear equations is recommended beforehand.

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Key points

This is the standard equation of a circle on a coordinate plane. Here, $(a, b)$ represents the coordinates of the center, and $r$ ( $r > 0$ ) is the radius.

(x - a)^2 + (y - b)^2 = r^2

This is the general form of a circle's equation. By completing the square for both $x$ and $y$ , you can convert it to standard form and determine the center and radius.

x^2 + y^2 + lx + my + n = 0

This formula determines the equation of the tangent line to the circle $x^2+y^2=r^2$ at a boundary point $(x_1, y_1)$ . Substituting the coordinates of the point of tangency directly yields the line equation.

x_1 x + y_1 y = r^2 \quad (x_1^2 + y_1^2 = r^2)

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