Pre-Calculus Prep: Conic Sections - Graph the Ellipse x^2 + 9y^2 + 6x -90y + 225 = 0

To graph the ellipse x^2 + 9y^2 + 6x - 90y + 225 = 0 and label the center, vertices, and co-vertices, follow these steps:

1. Rearrange the equation: Begin by rearranging the equation to put it in standard form. Group the x-terms and y-terms separately:
x^2 + 6x + 9y^2 - 90y + 225 = 0

2. Complete the square for the x-terms: Take half of the coefficient of x (which is 6) and square it, then add the result inside the parentheses. To maintain the equation's balance, you need to subtract the same value outside the parentheses:
(x^2 + 6x + 9) + 9y^2 - 90y + 225 = 9

3. Complete the square for the y-terms: Take half of the coefficient of y (which is -90) and square it, then add the result inside the parentheses. To maintain the equation's balance, you need to subtract the same value outside the parentheses:
(x^2 + 6x + 9) + 9(y^2 - 10y + 25) + 225 = 9 + 225
(x + 3)^2 + 9(y - 5)^2 = 234

4. Divide by the constant term: Divide both sides of the equation by the constant term (234 in this case) to make the right side equal to 1:
(x + 3)^2/234 + 9(y - 5)^2/234 = 1

5. Identify the center: The center of the ellipse is the opposite of the values inside the parentheses. In this case, the center is (-3, 5).

6. Determine the vertices: The vertices of the ellipse are the points where the major axis intersects the ellipse. For the equation (x + 3)^2/234 + 9(y - 5)^2/234 = 1, the vertices occur at (h ± a, k), where (h, k) represents the center and a is the length of the semi-major axis. The semi-major axis length can be found by taking the square root of the denominator of the x-term coefficient (in this case, √234).

- The vertices are at (-3 ± √234, 5).

7. Determine the co-vertices: The co-vertices are the points where the minor axis intersects the ellipse. For the equation (x + 3)^2/234 + 9(y - 5)^2/234 = 1, the co-vertices occur at (h, k ± b), where (h, k) represents the center and b is the length of the semi-minor axis. The semi-minor axis length can be found by taking the square root of the denominator of the y-term coefficient (in this case, √234/3).

- The co-vertices are at (-3, 5 ± √(234/3)).

8. Plot the center, vertices, and co-vertices: On a coordinate plane, plot the center at (-3, 5), the vertices at (-3 ± √234, 5), and the co-vertices at (-3, 5 ± √(234/3)).

9. Draw the ellipse: Connect the plotted points smoothly to form the elliptical shape of the graph.

By following these steps, you can graph the ellipse x^2 + 9y^2 + 6x - 90y + 225 = 0, and label the center, vertices, and co-vertices on a coordinate plane in pre-calculus.

These videos are designed to review and reteach Precalculus and Collegeboard Pre-CALC AP content. My videos cover functions, polynomials, exponential and logarithmic expressions, trigonometry, parametric equations, polar coordinates, vectors, matrices and systems, conic sections, discrete mathematics, sequences and series; and an introduction to calculus.

Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa .
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