Equation of a Circle
A circle of centre (h, k) and radius r has equation (x-h)2 + (y-k)2 = r2, or x2 + y2 + 2gx + 2fy + c = 0 in general form.
What is Equation of a Circle?
A circle of centre (h, k) and radius r has equation (x-h)2 + (y-k)2 = r2, or x2 + y2 + 2gx + 2fy + c = 0 in general form.
Key formula / rule: Centre is (-g, -f) and radius is √(g2 + f2 - c).
Key points
- Derive the centre and radius from the general equation of a circle.
- Compute the equation of a circle from stated geometric conditions.
Common exam trap
Reading the centre as (g, f) instead of (-g, -f).
Definitions
- Term
Equation of a Circle
- Meaning
A circle of centre (h, k) and radius r has equation (x-h)2 + (y-k)2 = r2, or x2 + y2 + 2gx + 2fy + c = 0 in general form.
- Term
Equation of a Circle — explanation
- Meaning
Completing the square converts the general form to the centre-radius form, revealing that the radius is real only when g2 + f2 - c is positive. That condition is frequently the examined point.
Learning objectives
Derive the centre and radius from the general equation of a circle.
Compute the equation of a circle from stated geometric conditions.
Formulae
- Key point
Centre is (-g, -f) and radius is √(g2 + f2 - c).
- Key point
A real circle requires g2 + f2 - c > 0.
- Key point
Three non-collinear points determine a unique circle.
Prerequisites
MAT-U3-SL-T1-S1-C1
Common mistakes
Reading the centre as (g, f) instead of (-g, -f).
Accepting a negative value under the radius square root.
Keywords
Equation
Circle
Practice preview
Find the equation of the circle with center (3, -2) and radius 5.…
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What are the coordinates of the center and the radius of the circle given by the equation (x+1)^2 + (y-4)^2 = 36?…
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Which of the following equations represents the general form of a circle?…
easy