Convert equations of circles from general to standard form

Learning Objectives Identify the general and standard forms of a circle's equation. Master the algebraic technique of 'completing the square' for quadratic expressions. Apply the completing the square method to convert a circle's equation from general form to standard form. Determine the center (h, k) and radius (r) of a circle from its general form equation. Analyze the resulting standard form to determine if the equation represents a circle, a point, or is degenerate. Verify a conversion by expanding the standard form equation back to its general form. Ever seen a jumble of x's, y's, and numbers and wondered if it's hiding a perfect circle? 🧐 Let's learn the secret to revealing it! In this tutorial, we will master a powerful algebr...

TermDefinitionExample Standard Form of a Circle's EquationThe form (x - h)^2 + (y - k)^2 = r^2, which explicitly reveals the circle's center (h, k) and its radius r.The equation (x - 5)^2 + (y + 1)^2 = 49 represents a circle with its center at (5, -1) and a radius of 7. General Form of a Circle's EquationThe form x^2 + y^2 + Dx + Ey + F = 0, where D, E, and F are constants. This form obscures the circle's center and radius.The equation x^2 + y^2 - 10x + 2y - 23 = 0 is in general form. Completing the SquareAn algebraic method used to convert a quadratic expression of the form x^2 + bx into a perfect square trinomial by adding the value (b/2)^2.To complete the square for x^2 + 12x, we add (12/2)^2 = 6^2 = 36. This gives x^2 + 12x + 36, which factors to (x + 6)^2. Perfect...

Standard Form Equation (x - h)^2 + (y - k)^2 = r^2 This is our target form. It directly provides the center (h, k) and the radius r. Remember that the value on the right side is r-squared. General Form Equation x^2 + y^2 + Dx + Ey + F = 0 This is our starting form. The circle's properties are not immediately obvious and must be found through conversion. The 'Completing the Square' Term For x^2 + bx, add (b/2)^2 This is the core calculation. To turn an expression like x^2 + bx into a perfect square, find half of the x-coefficient (b), square it, and add the result to both sides of your equation.