Equation of a Circle Formula

The Formula

(x - h)^2 + (y - k)^2 = r^2
General form: x^2 + y^2 + Dx + Ey + F = 0 (complete the square to convert to standard form).

When to use: A circle is the set of all points at the same distance (the radius) from a center point. The equation just says 'the distance from (x, y) to the center (h, k) equals r,' using the distance formula squared.

Quick Example

(x - 3)^2 + (y + 1)^2 = 25 This is a circle with center (3, -1) and radius r = 5.

Notation

Center (h, k), radius r. Note the signs: (x - h) means the center's x-coordinate is +h.

What This Formula Means

The standard form equation (x - h)^2 + (y - k)^2 = r^2 describes a circle with center (h, k) and radius r in the coordinate plane.

A circle is the set of all points at the same distance (the radius) from a center point. The equation just says 'the distance from (x, y) to the center (h, k) equals r,' using the distance formula squared.

Formal View

\{(x,y) \in \mathbb{R}^2 \mid (x-h)^2 + (y-k)^2 = r^2\}: the locus of points at distance r from center (h,k)

Worked Examples

Example 1

easy
Write the equation of the circle with center (3, -2) and radius 5.

Solution

  1. 1
    The standard form of a circle's equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) is the center and r is the radius.
  2. 2
    Substitute h = 3, k = -2, r = 5.
  3. 3
    (x - 3)^2 + (y + 2)^2 = 25.

Answer

(x - 3)^2 + (y + 2)^2 = 25
The equation of a circle is derived from the distance formula: every point (x, y) on the circle is exactly r units from the center (h, k). This gives \sqrt{(x-h)^2 + (y-k)^2} = r, which when squared yields the standard form.

Example 2

medium
Find the center and radius of the circle x^2 + y^2 - 6x + 4y - 12 = 0.

Common Mistakes

  • Sign errors with the center: (x - 3)^2 + (y + 1)^2 = 25 has center (3, -1), NOT (3, 1). The center coordinates are opposite the signs in the equation.
  • Forgetting to square the radius: if the radius is 5, the equation has r^2 = 25, not r = 5 on the right side.
  • Errors when completing the square: when adding (D/2)^2 to one side to complete the square, you must add the same value to the other side to keep the equation balanced.

Why This Formula Matters

Circles are the simplest conic section and the starting point for understanding ellipses, hyperbolas, and more advanced curves. They appear in physics (orbits, waves), engineering (gears, pipes), and computer graphics.

Frequently Asked Questions

What is the Equation of a Circle formula?

The standard form equation (x - h)^2 + (y - k)^2 = r^2 describes a circle with center (h, k) and radius r in the coordinate plane.

How do you use the Equation of a Circle formula?

A circle is the set of all points at the same distance (the radius) from a center point. The equation just says 'the distance from (x, y) to the center (h, k) equals r,' using the distance formula squared.

What do the symbols mean in the Equation of a Circle formula?

Center (h, k), radius r. Note the signs: (x - h) means the center's x-coordinate is +h.

Why is the Equation of a Circle formula important in Math?

Circles are the simplest conic section and the starting point for understanding ellipses, hyperbolas, and more advanced curves. They appear in physics (orbits, waves), engineering (gears, pipes), and computer graphics.

What do students get wrong about Equation of a Circle?

When converting from general form x^2 + y^2 + Dx + Ey + F = 0, complete the square for both x and y terms separately. Don't forget to add the same constants to both sides.

What should I learn before the Equation of a Circle formula?

Before studying the Equation of a Circle formula, you should understand: pythagorean theorem, domain.

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