Various forms of Circle Equation
Circle passing through 3 points:
1. Let x² + y² + 2gx + 2fy + c = 0 be circle passing through 3 points (non-collinear) then P (x₁, y₁), Q (x₂, y₂), R (x₃, y₃).
x₁² + y₁² + 2gx₁ + 2fy₁ + c = 0
x₂² + y₂² + 2gx₂ + 2fy₂ + c = 0
x₃² + y₃² + 2gx₃ + 2fy₃ + c = 0
Then equation of circle is \(\left| \begin{matrix}{{x}^{2}}+{{y}^{2}} & x & y & 1 \\x_{1}^{2}+y_{1}^{2} & {{x}_{1}} & {{y}_{1}} & 1 \\x_{2}^{2}+y_{2}^{2} & {{x}_{2}} & {{y}_{2}} & 1 \\x_{3}^{2}+y_{3}^{2} & {{x}_{3}} & {{y}_{3}} & 1 \\\end{matrix} \right|=0\)
2. If P is a point and C is the centre of circle of radius r, then maximum and minimum distances of P from the circle are CP + r and CP – r respectively.
If Lᵣ = aᵣx + bᵣy + cᵣ = 0, r = 1, 2, 3 be the sides of a triangle ABC then the equation of its circumcircle is \(\left| \begin{matrix}\frac{1}{{{L}_{1}}} &\frac{1}{{{L}_{2}}} & \frac{1}{{{L}_{3}}} \\ {{a}_{2}}{{a}_{3}}-{{b}_{2}}{{b}_{3}} & {{a}_{3}}{{a}_{1}}-{{b}_{3}}{{b}_{1}} & {{a}_{1}}{{a}_{2}}-{{b}_{1}}{{b}_{2}} \\ {{a}_{2}}{{b}_{3}}+{{a}_{3}}{{b}_{2}} & {{a}_{3}}{{b}_{1}}+{{a}_{1}}{{b}_{3}} & {{a}_{1}}{{b}_{2}}+{{a}_{2}}{{b}_{1}} \\\end{matrix} \right|=0\)
3. The general equation of circle passing through two given points A (x₁, y₁), B (x₂, y₂) may be written as (x – x₁) (x – x₂) + (y – y₁) (y – y₂) + λ \(\left| \begin{matrix}x & y & 1 \\{{x}_{1}} & {{y}_{1}} & 1 \\{{x}_{2}} & {{y}_{2}} & 1 \\\end{matrix} \right|\) = 0 where λ ϵ R.
4. Diameter from of circle: The equation of circle drawn on the straight-line segment joining points A (x₁, y₁), B (x₂, y₂) as diameter is (x – x₁) (x – x₂) + (y – y₁) (y – y₂) = 0
5. Equation of circle in parametric form: Parametric equations of x² + y² = r²: here center is origin x = r cosθ, y = r sinθ be any point on circle.
∴ Parametric equations are x = r cosθ and y = r sinθ.
Parametric equations of (x – a)² + (y – b)² = r² is x = a + r cosθ, y = b + r sinθ.