Cartesian equation of a circle — A/L Combined Mathematics

What must I remember about Cartesian equation of a circle?

• Definition of a Circle: Locus of a point P(x, y) moving in a plane such that its distance from a fixed point (center C) remains constant (radius r > 0).
• Standard Central Form: Circle with center C(h, k) and radius r has equation (x - h)² + (y - k)² = r²; circle centered at origin (0, 0) has x² + y² = r².
• General Form of Circle: x² + y² + 2gx + 2fy + c = 0, where Center is (-g, -f) and Radius is r = √(g² + f² - c); requires coefficients of x² and y² to be equal and non-zero, with zero xy cross-term.
• Nature of Circle: Real circle if g² + f² - c > 0; Point circle (single real point (-g, -f)) if g² + f² - c = 0; Imaginary / Virtual circle (no real locus) if g² + f² - c < 0.
• Diameter form of Circle: Circle having points A(x1, y1) and B(x2, y2) as endpoints of a diameter has equation (x - x1)(x - x2) + (y - y1)(y2 - y) = 0.

Test yourself (1 minute)

1. Circle through three non-collinear points

   Substitute (x1, y1), (x2, y2), (x3, y3) into x² + y² + 2gx + 2fy + c = 0 to obtain three linear equations in parameters g, f, c, and solve simultaneously.

2. Point position relative to a circle

   For point P(x1, y1) and S ≡ x² + y² + 2gx + 2fy + c = 0: P lies outside the circle if S1 > 0; P lies on the circle if S1 = 0; P lies inside the circle if S1 < 0 (where S1 = x1² + y1² + 2gx1 + 2fy1 + c).

3. Intercepts made on coordinate axes

   Circle x² + y² + 2gx + 2fy + c = 0 intercepts length 2√(g² - c) on the x-axis, and length 2√(f² - c) on the y-axis; tangent to x-axis if g² = c, tangent to y-axis if f² = c.

Practise Cartesian equation of a circle questions today

Source: NIE G.C.E. (A/L) Combined Mathematics Resource Book - Pure Mathematics, Unit 19

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Year 1 · Term 3