Equation of a Circle

1Summary 重點摘要

Standard and general circle equation, find center and radius from equation.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Standard form — read off the centre and radiusThe standard form of a circle is (x - h)² + (y - k)² = r² with centre (h, k) and radius r; it follows directly from the distance formula, since every point on the circle is exactly r from the centre. The brackets always subtract: (x - 3)² + (y + 2)² = 16 has centre (3, -2) and radius 4, because y + 2 means y - (-2). Two classic slips: reading the centre as (3, 2), and forgetting that the right side is r², so a square root is needed before the radius appears.
2. Step 2: General form — reading the three coefficientsThe general form is x² + y² + Dx + Ey + F = 0, recognised by the equal coefficients of x² and y² and the absence of an xy term. It hides the geometry, so rely on two formulas: the centre is (-D2, -E2) and the radius is √[(D2)² + (E2)² - F]. Watch the signs — the centre takes a minus sign in each coordinate, and F enters the radius formula with a minus. If the expression under the root turns out negative, the equation represents no real circle, so check it before answering.
3. Step 3: Completing the square — back to standard formCompleting the square converts the general form to the standard form: write x² + Dx as (x + D2)² - (D2)², treat y² + Ey the same way, move every constant to the right, and the centre and radius can be read off. The method matters because the standard form displays the geometry while the general form does not. A common error is dropping the subtracted (D2)² term when expanding, which inflates r²; tidy the fractional coefficients before squaring so no sign error creeps into the working.
4. Step 4: Building the equation from geometric conditionsMany questions run the other way: given geometric conditions, find the equation. Pin down the centre and the radius first. With the two ends of a diameter, use the midpoint formula for the centre and the distance formula for the radius. With the centre and one point on the circle, the distance between them is the radius. When the circle touches the x-axis, the radius equals the absolute value of the y-coordinate of the centre. Then substitute both into the standard form. Beware the tangent condition when the centre sits on a negative axis — skipping the absolute value produces a negative radius.
5. Step 5: Position of a point relative to the circleTo locate a point P(x₀, y₀) relative to the circle, substitute into the left side and compare with the right: if (x₀ - h)² + (y₀ - k)² is smaller than r², the point is inside; equal means on the circle; larger means outside. The reason is that the left side is the squared distance from P to the centre, so the comparison is really between a distance and the radius. Such tests often appear with tangent and intersection questions. Do not compare a single coordinate without squaring and adding, and do not swap the inside and outside conditions — inside simply means closer to the centre.
6. Step 6: Verification and the common question typesAfter finding the equation, run two quick checks: the distance from the centre to any point on the circle equals the radius, and x² and y² share the same coefficient with no xy term. Frequent tasks include extracting the centre and radius from the general form, building the equation from geometric conditions, classifying a point's position, and intersecting a circle with a line — all revolving around the centre and the radius. Do not stop verifying after a single number; pick a point that is easy to compute, such as one level horizontally or vertically with the centre.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Standard: (x-h)² + (y-k)² = r²
• Center (h, k), radius r
• (x - h)² center h no need flip
• Right side is r², must square root get radius
• General form center (-D2, -E2)