Question

Hi! I have a question and I don't understand it's answer. Could you help me?It's in the attachment below.

Answer 1

To solve this question, we just need to evaluate our set of points in the standard form equation of a Hyperbola, and find the coefficients. This will give to us the equation for our Hyperbola. The standard form is

$\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$

Let's start with the easier points, the x-intercepts (5, 0) and (-1, 0).

Since this hyperbola has two x-intercepts, we're dealing with a horizontal hyperbola, and the center is the midpoint between the x-intercepts.

$\begin{gathered} \bar{x}=\frac{x_1+x_2}{2}=\frac{-1+5}{2}=2 \\ \bar{y}=\frac{y_1+y_2}{2}=\frac{0+0}{2}=0 \end{gathered}$

The center coordinates are (2, 0), then, our equation is

$\frac{(x-2)^2}{a^2}-\frac{y^2}{b^2}=1$

To find the missing coefficients, we can just substitute the remaining points and solve the system for a and b. Our final equation is

$\frac{(x-2)^2}{9^{}}-\frac{y^2}{4}^{}=1$