Find properties of circles from equations in general form

Learning Objectives Identify the general form of a circle's equation. Convert the general form of a circle's equation to its standard form using the method of completing the square. Determine the coordinates of the center (h, k) of a circle from its equation. Calculate the radius (r) of a circle from its equation. Analyze the equation to determine if it represents a circle, a single point (a point circle), or is degenerate (no graph). Solve problems involving finding properties of circles from equations in general form. Ever seen a GPS pinpoint your location within a certain radius? 🗺️ That's a circle in action! But how do we find that circle's center and range from a complex-looking equation? This tutorial will teach you how to take the 'general fo...

TermDefinitionExample Standard Form of a CircleThe equation of a circle written as (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius. This form makes the circle's properties immediately obvious.The equation (x - 2)² + (y + 5)² = 16 represents a circle with its center at (2, -5) and a radius of 4. General Form of a CircleThe equation of a circle written as x² + y² + Dx + Ey + F = 0, where D, E, and F are constants. The center and radius are not immediately apparent from this form.The equation x² + y² - 4x + 10y + 13 = 0 is in general form. Its properties are hidden and must be found through conversion. Center of a Circle (h, k)The fixed point in the middle of a circle from which all points on the circumference are equidistant.For a circle with equation (x - 3)²...

Standard Form Equation (x - h)^2 + (y - k)^2 = r^2 Use this form to identify the center (h, k) and the radius r. Remember to be careful with the signs of h and k. General Form Equation x^2 + y^2 + Dx + Ey + F = 0 This is the form you will typically start with. To find the properties, you must convert it to standard form. Note that the coefficients of x² and y² must be 1 for this form. Conversion Formulas (from General Form) h = -D/2, \quad k = -E/2, \quad r = \sqrt{h^2 + k^2 - F} These are shortcut formulas to find the center (h, k) and radius r directly from the coefficients D, E, and F in the general form. The method of completing the square is the process behind these formulas.