Question

find an equation for the hyperbola described. graph the equation
center at (0,0), focus at (4,0), vertex at (1,0)
an equation of the hyperbola is □=1.
(use integers or fractions for any numbers in the expression.)
choose the correct graph below.

Explanation:

Step1: Recall the standard form of a hyperbola

The standard form of a hyperbola centered at \((h,k)\) with a horizontal transverse axis is \(\frac{(x - h)^2}{a^2}-\frac{(y - k)^2}{b^2}=1\). Here, \(h = 0,k = 0\) (center \((0,0)\)), \(a\) is the distance from the center to the vertex. Given vertex \((1,0)\), so \(a = 1\) (since \(|x - h|=|x-0| = 1\) at the vertex). The distance from the center to the focus is \(c\). Given focus \((4,0)\), so \(c = 4\).

Step2: Use the relationship \(c^2=a^2 + b^2\)

We know that \(c^2=a^2 + b^2\). Substituting \(a = 1\) and \(c = 4\) into the formula: \(4^2=1^2 + b^2\). Then \(b^2=c^2 - a^2=16 - 1=15\).

Step3: Write the equation of the hyperbola

Substituting \(h = 0,k = 0,a^2 = 1,b^2 = 15\) into the standard form \(\frac{(x - h)^2}{a^2}-\frac{(y - k)^2}{b^2}=1\), we get \(\frac{x^{2}}{1}-\frac{y^{2}}{15}=1\).

Answer:

\(\frac{x^{2}}{1}-\frac{y^{2}}{15}=1\)