Circles

4

Given equation: x^2 + y^2 - 6x + 8y + 9 = 0. Comparing with general form, 2g = -6 ⇒ g = -3, 2f = 8 ⇒ f = 4, c = 9. Radius = √(g² + f² - c) = √(9 + 16 - 9) = √16 = 4.

x^2 + y^2 + 4x - 6y + 9 = 0,2x^2 + 2y^2 - 8x + 12y + 18 = 0

For an equation to represent a circle, coefficients of x² and y² must be equal and non-zero, and there should be no xy term. Both first and fourth equations satisfy this after simplifying the fourth by dividing by 2.

(x - 1)(x - 3) + (y - 2)(y - 4) = 0

The diameter form of a circle is (x - x₁)(x - x₂) + (y - y₁)(y - y₂) = 0. Substituting (1,2) and (3,4) gives the correct equation.

x = 1 + 3cosθ, y = -2 + 3sinθ

For a circle (x - h)^2 + (y - k)^2 = r^2, parametric form is x = h + r cosθ, y = k + r sinθ. Here h = 1, k = -2, r = 3.