9.2 Standard and General Equations of Circles

9.2 Standard and General Equations of Circles

A circle is the geometric locus of all points in a plane that are at a fixed distance (the radius $r > 0$) from a fixed point (the center $C(a, b)$). In Gaokao analytic geometry, questions on circles test structural algebraic transformations, geometric properties (such as chord length, tangency, and symmetry), parametric optimizations, locus derivations, and interactions between multiple circles.

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1. Standard and General Forms of Circle Equations

Standard Form

Applying the distance formula between a dynamic point $P(x, y)$ and fixed center $C(a, b)$ set equal to radius $r > 0$:

$$\sqrt{(x - a)^2 + (y - b)^2} = r \implies (x - a)^2 + (y - b)^2 = r^2$$

• Key Advantage: Standard Form explicitly reveals the center coordinates $(a, b)$ and radius $r$. It is preferred when geometric conditions involve center coordinates or radius explicitly.

General Form

Expanding the standard form equation:

$$x^2 - 2ax + a^2 + y^2 - 2by + b^2 - r^2 = 0$$

Letting $D = -2a$, $E = -2b$, and $F = a^2 + b^2 - r^2$, we obtain the General Form:

$$x^2 + y^2 + Dx + Ey + F = 0$$

To recover geometric parameters from the general form, complete the square for $x$ and $y$:

$$\left(x + \frac{D}{2}\right)^2 + \left(y + \frac{E}{2}\right)^2 = \frac{D^2 + E^2 - 4F}{4}$$

Discriminant Analysis of General Form

The equation $x^2 + y^2 + Dx + Ey + F = 0$ is governed by the discriminant $\Delta_{circle} = D^2 + E^2 - 4F$:

1. $\Delta_{circle} > 0$: Represents a real circle with center $C\left(-\frac{D}{2}, -\frac{E}{2}\right)$ and radius $r = \frac{1}{2}\sqrt{D^2 + E^2 - 4F}$.
2. $\Delta_{circle} = 0$: Represents a single point $C\left(-\frac{D}{2}, -\frac{E}{2}\right)$ (a degenerate circle with radius $r = 0$).
3. $\Delta_{circle} < 0$: Represents no real locus (empty set in $\mathbb{R}^2$).

Gaokao Caution: Never assume an equation of the form $x^2 + y^2 + Dx + Ey + F = 0$ represents a circle without first checking or enforcing $D^2 + E^2 - 4F > 0$.

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2. Parametric Equations of a Circle

For a circle $(x - a)^2 + (y - b)^2 = r^2$, introducing parameter angle $\theta \in [0, 2\pi)$ (measured counterclockwise from the positive horizontal direction centered at $C(a, b)$) gives the parametric equations:

$$\begin{cases} x = a + r \cos \theta \\ y = b + r \sin \theta \end{cases} \quad (\theta \in [0, 2\pi))$$

Applications of Parametric Equations

• Algebraic Extremum Problems: To find the maximum or minimum of a linear objective function $z = Ax + By$ for points $(x, y)$ on circle $(x - a)^2 + (y - b)^2 = r^2$, substitute parametric coordinates: $$z = A(a + r\cos\theta) + B(b + r\sin\theta) = (Aa + Bb) + r(A\cos\theta + B\sin\theta) = (Aa + Bb) + r\sqrt{A^2 + B^2}\sin(\theta + \phi)$$ where $\tan\phi = \frac{A}{B}$. Thus $z_{\max} = Aa + Bb + r\sqrt{A^2 + B^2}$ and $z_{\min} = Aa + Bb - r\sqrt{A^2 + B^2}$.

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3. Methods for Determining Equations of Circles

Determining a circle's equation requires finding three independent algebraic coefficients (either $a, b, r$ or $D, E, F$). Two principal methods exist:

Algebraic Method (Undetermined Coefficients)

1. Assume general form $x^2 + y^2 + Dx + Ey + F = 0$.
2. Set up a system of three linear equations using three given non-collinear points $P_1, P_2, P_3$.
3. Solve for $D, E, F$ and verify $D^2 + E^2 - 4F > 0$.

Geometric Method

1. Use geometric properties to locate center $C(a, b)$:
   • Center $C$ lies on the perpendicular bisectors of any chords formed by given points.
   • Center $C$ lies at distance $r$ from tangent lines.
2. Compute radius $r = |CP_1|$.
3. Write the standard form $(x - a)^2 + (y - b)^2 = r^2$.

Diameter Form of a Circle

If segment $AB$ with endpoints $A(x_1, y_1)$ and $B(x_2, y_2)$ is a diameter of circle $C$, then any point $P(x, y)$ on the circle (except $A, B$) satisfies $\vec{PA} \perp \vec{PB} \iff \vec{PA} \cdot \vec{PB} = 0$. Expanded equation:

$$(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0$$

Apollonius Circle (阿波罗尼斯圆)

In the plane, the locus of point $P$ such that the ratio of its distances to two fixed points $A$ and $B$ is a constant $k$ ($k > 0, k \neq 1$) is a circle:

$$\frac{|PA|}{|PB|} = k \implies |PA|^2 = k^2 |PB|^2$$

Derivation: Place $A(-c, 0)$ and $B(c, 0)$ on the $x$-axis. Expanding $(x + c)^2 + y^2 = k^2 [(x - c)^2 + y^2]$ yields a quadratic equation with equal non-zero coefficients for $x^2$ and $y^2$ and no $xy$ term, representing a circle.

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4. Positional Relationships Between Two Circles

Let two circles be $C_1: (x - a_1)^2 + (y - b_1)^2 = R^2$ and $C_2: (x - a_2)^2 + (y - b_2)^2 = r^2$ with $R \ge r > 0$. Let $d = |C_1 C_2| = \sqrt{(a_1 - a_2)^2 + (b_1 - b_2)^2}$ be the center-to-center distance.

Common Chord and Radical Axis (相交弦与等幂轴)

When two circles $C_1: x^2 + y^2 + D_1 x + E_1 y + F_1 = 0$ and $C_2: x^2 + y^2 + D_2 x + E_2 y + F_2 = 0$ intersect at two points $A$ and $B$, subtracting $C_2$ from $C_1$ eliminates $x^2$ and $y^2$:

$$(D_1 - D_2)x + (E_1 - E_2)y + (F_1 - F_2) = 0$$

This linear equation represents the radical axis (等幂轴) of the two circles. For intersecting circles, it is precisely the line equation of their common chord $AB$.

• Common Chord Length Formula: Perpendicular distance $d_1$ from center $C_1$ to line $AB$ gives chord length: $$L_{AB} = 2 \sqrt{R^2 - d_1^2}$$