Question

consider the following hyperbola.

$$ \frac { ( x + 8 ) ^ { 2 } } { 4 } - \frac { ( y + 4 ) ^ { 2 } } { 25 } = 1 $$

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

Explanation:

Step1: Identify the form of the hyperbola

The standard form of a hyperbola is \(\frac{(x - h)^2}{a^2}-\frac{(y - k)^2}{b^2}=1\). For the given hyperbola \(\frac{(x + 8)^2}{4}-\frac{(y + 4)^2}{25}=1\), we have \(h=-8\), \(k = - 4\), \(a^2=4\) (so \(a = 2\)), \(b^2=25\) (so \(b = 5\)).

Step2: Calculate \(c\)

For a hyperbola, the relationship \(c^2=a^2 + b^2\) holds. Substitute \(a = 2\) and \(b = 5\) into the formula: \(c^2=4 + 25=29\), so \(c=\sqrt{29}\).

Step3: Find the foci coordinates

Since the hyperbola is of the form \(\frac{(x - h)^2}{a^2}-\frac{(y - k)^2}{b^2}=1\) (opens left - right), the foci are at \((h\pm c,k)\). Substitute \(h=-8\), \(k=-4\) and \(c = \sqrt{29}\) into the formula. The foci are \((-8+\sqrt{29},-4)\) and \((-8-\sqrt{29},-4)\).

Answer:

The coordinates of the foci are \((-8+\sqrt{29},-4)\) and \((-8-\sqrt{29},-4)\)