Question

A hyperbola centered at (2, 0) has a focus at (2, 10) and a vertex at (2, 8). What is the equation of the hyperbola in standard form? a. [(x-2)²]/100 - y²/64 = 1 b. y²/100 - (x-2)²]/64 = 1 c. [(x-2)²]/64 - y²/36 = 1 d. y²/64 - (x-2)²]/36= 1

Answer 1

The correct equation of the hyperbola in standard form is y²/64 - [(x-2)²]/36 = 1, which describes a vertical hyperbola centered at (2, 0) with a vertex at (2, 8) and a focus at (2, 10).

Step-by-step explanation:

The equation of a hyperbola can be written in standard form as [(x-h)²]/a² - [(y-k)²]/b² = 1 if the hyperbola is horizontal, or [(y-k)²]/a² - [(x-h)²]/b² = 1 if the hyperbola is vertical, where (h, k) is the center of the hyperbola, a is the distance from the center to a vertex, and b is the distance from the center to the co-vertex. In this case, since the center of the hyperbola is at (2, 0), and there is a vertex at (2, 8) along the vertical direction, the equation will have the vertical form. The distance from the center to the vertex (the value of a) is 8 units. The focus being at (2, 10) means that the distance from the center to the focus (the value of c) is 10 units. We can find b by using the relationship c² = a² + b², so b² = c² - a² = 10² - 8² = 36. Hence, the equation is y²/64 - [(x-2)²]/36 = 1, which corresponds to option d.

Answer 2

The equation of the hyperbola in standard form is [(x-2)²]/100 - y²/64 = 1.

Step-by-step explanation:

The equation of a hyperbola in standard form is given by (x - h)2/a2 - (y - k)2/b2 = 1, where (h, k) is the center, and 'a' and 'b' are the distances from the center to the vertices along the x and y axis respectively.

In this case, the hyperbola is centered at (2, 0), so we have h = 2 and k = 0. The distances from the center to the vertices along the x-axis and y-axis are 1 and 2 respectively.

Plugging these values into the standard form equation, we get (x - 2)2/12 - (y - 0)2/22 = 1. Simplifying further, we have (x - 2)2/1 - y2/4 = 1. Therefore, the equation of the hyperbola in standard form is option a, [(x-2)²]/100 - y²/64 = 1.