Find properties of ellipses from equations in general form

Learning Objectives Identify an equation in general form as an ellipse. Convert the general form of an ellipse's equation to its standard form by completing the square. Determine the coordinates of the center (h, k) and the orientation of the major axis. Calculate the lengths of the major and minor axes. Find the coordinates of the vertices and co-vertices. Calculate the focal distance 'c' and determine the coordinates of the foci. How can a messy equation like 4x² + 9y² - 16x + 18y - 11 = 0 describe the perfect, graceful orbit of a planet? 🪐 Let's find out! This tutorial will guide you through the process of taking a complicated-looking general form equation and transforming it into the much friendlier standard form of an ellipse. By doing this, you wi...

TermDefinitionExample General Form of an EllipseAn equation of the form Ax² + Cy² + Dx + Ey + F = 0, where A and C are positive and A ≠ C. This form describes a conic section, but it doesn't immediately reveal the ellipse's properties.9x² + 4y² + 36x - 24y + 36 = 0 Standard Form of an EllipseThe form of an ellipse's equation that clearly shows its properties. It is written as \frac{(x-h)²}{a²} + \frac{(y-k)²}{b²} = 1 or \frac{(x-h)²}{b²} + \frac{(y-k)²}{a²} = 1, where a > b.\frac{(x+2)²}{4} + \frac{(y-3)²}{9} = 1 Completing the SquareAn algebraic technique used to convert a quadratic expression from a general form (ax² + bx) to a perfect square trinomial ((x+d)²). This is the key method for converting the general form of an ellipse to its standard form.To complete the sq...

Conversion by Completing the Square 1. Group x-terms and y-terms. Move constant F to the right side. 2. Factor out coefficients A and C from the x and y groups respectively. 3. Complete the square for x and y inside the parentheses. 4. Add the corresponding values to the right side. 5. Divide the entire equation by the new constant on the right to make it equal to 1. This is the step-by-step process to transform the general form Ax² + Cy² + Dx + Ey + F = 0 into the standard form \frac{(x-h)²}{a²} + \frac{(y-k)²}{b²} = 1. Focal Length Formula c² = a² - b² Use this formula to find the distance 'c' from the center to each focus. Remember that a² is always the larger denominator in the standard form equation. Coordinates of Key Points Center: (h, k) Horizonta...