Question

consider the following hyperbola.

$$\frac{(x + 3)^{2}}{25}-\frac{(y + 5)^{2}}{64}=1$$

step 2 of 3: find the coordinates of the foci of the hyperbola.

Explanation:

Step1: Identify the form of hyperbola

The standard form of a hyperbola is \(\frac{(x - h)^2}{a^2}-\frac{(y - k)^2}{b^2}=1\) (horizontal transverse axis). Here \(h=-3,k = - 5,a^{2}=25\), so \(a = 5\), \(b^{2}=64\), so \(b = 8\).

Step2: Calculate \(c\)

For a hyperbola, \(c^{2}=a^{2}+b^{2}\). Substitute \(a = 5\) and \(b = 8\) into the formula: \(c^{2}=25 + 64=89\), then \(c=\sqrt{89}\).

Step3: Find the foci coordinates

Since the hyperbola has a horizontal transverse axis, the foci are \((h\pm c,k)\). Substitute \(h=-3,k=-5,c = \sqrt{89}\) into the formula. The foci are \((-3+\sqrt{89},-5)\) and \((-3-\sqrt{89},-5)\).

Answer:

The coordinates of the foci are \((-3+\sqrt{89},-5)\) and \((-3-\sqrt{89},-5)\)