Question

A hyperbola centered at (7, 0) has a focus at (7, 5) and vertex at (7, 4). Which is the equation of the hyperbola in standard form?

quantity x minus 7 end quantity squared over 16 minus y squared over 9 equals 1
quantity x minus 7 end quantity squared over 25 minus y squared over 16 equals 1
y squared over 16 minus quantity x minus 7 end quantity squared over 9 equals 1
y squared over 25 minus quantity x minus 7 end quantity squared over 16 equals 1

Answer 1

Based on the calculations, the equation of this hyperbola in standard form is: A. $\frac{x\;-\;7}{16} + \frac{y}{9} = 1$.

How to determine the equation of a hyperbola?

Mathematically, the equation of a hyperbola in standard form is given by:

$\frac{x\;-\;h}{a^2} + \frac{x\;-\;k}{b^2} = 1$

Given the following data:

Center (h, k) = (7, 0)

Vertex (h+a, k) = (7, 4)

Focus = (h+c, k) = (7, 5)

Also, we can deduce that the value of a and c are 4 and 5 respectively.

For the value of b, we would apply Pythagorean's theorem:

c² = a² + b²

b² = c² - a²

b² = 5² - 4²

b² = 9.

Substituting the parameters into the standard equation, we have:

$\frac{x\;-\;7}{4^2} + \frac{y\;-\;0}{3^2} = 1\\\\\frac{x\;-\;7}{16} + \frac{y}{9} = 1$

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