Question

Find an equation of the hyperbola. Given: Vertices (7, ±3), Foci (7, ±5).

a) (x - 7)²/9 - (y + 3)²/25 = 1
b) (x + 7)²/9 - (y - 3)²/25 = 1
c) (x - 7)²/25 - (y + 3)²/9 = 1
d) (x + 7)²/25 - (y - 3)²/9 = 1

Answer 1

The equation of the hyperbola with the given vertices and foci is (x - 7)²/9 - y²/25 = 1, which is option a).

Step-by-step explanation:

To find an equation of the hyperbola with the given vertices (7, ±3) and foci (7, ±5), we recall that the standard form of the equation of a hyperbola that opens up and down (since the vertices and foci have the same x-coordinates) is:

(x - h)²/a² - (y - k)²/b² = 1,

where (h, k) is the center of the hyperbola, a is the distance from the center to each vertex, and b is the distance from the center to each co-vertex.

Here, the center of the hyperbola is (7, 0), since it is midway between both the vertices and the foci. We can calculate the value of 'a' as the distance from the center to any vertex, which is 3. Thus, a² = 3² = 9.

The focal distance (the distance from the center to a focus) is 5, and for a hyperbola, c² = a² + b². Thus, 5² = 9 + b², giving us b² = 25.

Therefore, the equation of the hyperbola is:

(x - 7)²/9 - (y - 0)²/25 = 1, which simplifies to:

(x - 7)²/9 - y²/25 = 1

This matches option a).

Answer 2

The correct equation of the hyperbola is option c) (x - 7)²/25 - (y + 3)²/9 = 1.(option c)

Step-by-step explanation:

In a hyperbola, the standard form of the equation is given by (x - h)²/a² - (y - k)²/b² = 1, where (h, k) is the center, 2a is the length of the major axis, and 2b is the length of the minor axis.

Firstly, observe that the vertices are (7, ±3), and the foci are (7, ±5). This implies that the center of the hyperbola is at (7, 0) since the x-coordinate of the center remains the same, and the y-coordinate changes between the vertices and foci.

Next, determine a and b. The distance between the center and vertices is a = 5 - 0 = 5, and the distance between the center and foci is c = 5. Use the relationship c² = a² + b² for hyperbolas, where c is the distance from the center to the foci. Solve for b, which gives b = √(c² - a²) = √(25 - 9) = √16 = 4.

Now, substitute these values into the standard form, resulting in (x - 7)²/25 - (y + 3)²/9 = 1, matching option c).

Therefore, the correct equation for the hyperbola is (x - 7)²/25 - (y + 3)²/9 = 1. (option c)