Question

Find an equation for the conic that satisfies the given conditions. hyperbola, vertices (−2, −4), (−2, 6), foci (−2, −7), (−2, 9)

Answer 1

To find the equation of the hyperbola, use the standard form of the equation and substitute the given values of the vertices and foci. Equation comes to be
`(x + 2)^2 / 25 - (y - 1)^2 / 64 = 1`

Step-by-step explanation:

To find the equation of a hyperbola given the vertices and foci, we can use the standard form of the equation:

`(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1`

where (h, k) is the center of the hyperbola, and a and b represent the distance from the center to the vertices and from the center to the foci, respectively.

Given that the vertices are (-2, -4) and (-2, 6), we can determine that the center is (-2, 1) since it lies in the horizontal midline between the vertices. Moreover, the distance from the center to the vertices is 5, so a will be equal to 5. The distance from the center to the foci is 8, so b will be equal to 8.

Using these values, we can substitute into the standard form equation and get:

`(x + 2)^2 / 25 - (y - 1)^2 / 64 = 1`

Answer 2

To find the equation of the hyperbola, we need to determine the center, the distances from the center to the vertices and foci, and use the appropriate equation. The equation of the hyperbola is
`((x + 2)^2 / 5^2 - (y - 1)^2)/(39) = 1.`

To find the equation of a hyperbola given its vertices and foci, we need to determine the center and the distances from the center to the vertices and foci. In this case, we can see that the center of the hyperbola is the point (-2, 1), which is the midpoint between the given vertices. The distance from the center to the vertices is 5 units (6 - 1 = 5), and the distance from the center to the foci is 8 units (9 - 1 = 8).

The equation of a hyperbola centered at (h, k) with vertices a units away from the center and foci c units away from the center is given by:


`(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1`

Plugging in the values, we have:


`(x + 2)^2 / 5^2 - (y - 1)^2 / b^2 = 1\\`
Since we know the distance from the center to the foci (c = 8) and the distance from the center to the vertices (a = 5), we can use the relationship
`c^2 = a^2 + b^2` to solve for b. Solving for b, we get
`b^2 = c^2 - a^2 = 8^2 - 5^2 = 39.` Therefore, the equation of the hyperbola is:


`(x + 2)^2 / 5^2 - (y - 1)^2 / 39 = 1`