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The document discusses the standard equation formulas for ellipses depending on the orientation of the major axis. It provides two standard forms: one for when the major axis is horizontal with x2/a2 + y2/b2 = 1, and one for when the major axis is vertical with x2/b2 + y2/a2 = 1. It defines the key terms used in the equations like center, foci, vertices and co-vertices. Examples are given to demonstrate how to identify which equation form to use based on the values in the denominators.
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Moudle#7 notes
- 1. Mathematical Key Concepts… (1) A Mathematical Key Points… The standard equation formula of the ellipse has two standard equation formulas depending on its major axis. HORIZONTAL AMJOR AXIS VERTICAL MAJOR AXIS 𝑥 − ℎ 2 𝑎2 + 𝑦 − 𝑘 2 𝑏2 = 1 𝑥 − ℎ 2 20 + 𝑦 − 𝑘 2 10 = 1 Center ℎ, 𝑘 Length of the major axis 2𝑎 Length of the minor axis 2𝑏 Distance between the center and either the focus: 𝑐2 = 𝑎2 − 𝑏2 Vertices ℎ ± 𝑎, 𝑘 Foci ℎ ± 𝑐, 𝑘 Co-vertices ℎ, 𝑘 ± 𝑏 NOTE: You will be using this formula if the bigger value in the denominator is on the first term Example: 𝑥 − ℎ 2 𝑏2 + 𝑦 − 𝑘 2 𝑎2 = 1 𝑥 − ℎ 2 20 + 𝑦 − 𝑘 2 30 = 1 Center ℎ, 𝑘 Length of the major axis 2𝑎 Length of the minor axis 2𝑏 Distance between the center and either the focus: 𝑐2 = 𝑎2 − 𝑏2 Vertices ℎ, 𝑘 ± 𝑏 Foci ℎ, 𝑘 ± 𝑐 Co-vertices ℎ ± 𝑏, 𝑘 NOTE: You will be using this formula if the bigger value in the denominator is on the second term Example:
- 2. EXAMPLES (1) Solve the standard equation form of an ellipse. a. + − + = SOLUTION: Explanation + − + = *Given − + + = *Arrange the terms according to and coefficients − + + = *Factoring the terms − + + + + = + + *Completing the perfect square trinomial − + + = *Factor Perfect Trinomial Square − + + = *Divide both sides by 576 − + + = *The standard equation form of an ellipse *Since the bigger value on the denominator can be found on the first term then its major axis is horizontal and we will use the formulas:
- 3. − ℎ 2 2 + − 2 2 = 1