Question

(1 point) Determine the distance D between the vertices of

-9x² + 18x + 4y² + 24y - 9 = 0.
D =

Answer 1

To find the distance between the vertices of the ellipse given by the equation -9x² + 18x + 4y² + 24y - 9 = 0, we first need to rewrite the equation in standard form by completing the square. Then, we can find the distance between the vertices by identifying the value of 'a' in the standard form equation and doubling it. In this case, the distance between the vertices is 6 units.

Step-by-step explanation:

To determine the distance between the vertices of the equation -9x² + 18x + 4y² + 24y - 9 = 0, we need to find the distance between two points in the coordinate plane. The given equation represents an ellipse, so we can use the distance formula. First, we need to rewrite the equation in standard form by completing the square for both x and y. Then, we can find the distance between the vertices by finding the difference in x-coordinates and y-coordinates. Let's go through the steps:

1. Start with the equation -9x² + 18x + 4y² + 24y - 9 = 0
2. Complete the square for x: -9(x² - 2x) + 4y² + 24y - 9 = 0. Rewrite the x-term as (x - 1)² to complete the square: -9(x - 1)² + 4y² + 24y - 9 = 0.
3. Complete the square for y: -9(x - 1)² + 4(y² + 6y) - 9 = 0. Rewrite the y-term as (y + 3)² to complete the square: -9(x - 1)² + 4(y + 3)² - 9 = 0.
4. Now the equation is in standard form: -9(x - 1)² + 4(y + 3)² = 9.
5. Identify the vertex coordinates: The vertex coordinates are (1, -3).
6. Identify the distance between the vertices: The distance between the vertices is twice the value of 'a' in the standard form equation. In this case, 'a' = 3. Therefore, the distance between the vertices is 2 * 3 = 6 units.