Questions: Find the vertices and locate the foci of the hyperbola with the given equation. Then graph the equation. x^2/25 - y^2/4 = 1 The vertices of the hyperbola are (Type an ordered pair. Simplify your answer. Use a comma to separate answers as needed.)

Solution Steps

The given equation is: \[ \frac{x^{2}}{25}-\frac{y^{2}}{4}=1 \] This is in the standard form of a hyperbola: \[ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \] where \(a^{2} = 25\) and \(b^{2} = 4\).

The vertices of the hyperbola are located at \((\pm a, 0)\). Given \(a^{2} = 25\), we find \(a = \sqrt{25} = 5\). Thus, the vertices are: \[ (5, 0) \text{ and } (-5, 0) \]

The foci of the hyperbola are located at \((\pm c, 0)\), where \(c = \sqrt{a^{2} + b^{2}}\). Given \(a^{2} = 25\) and \(b^{2} = 4\), we find: \[ c = \sqrt{25 + 4} = \sqrt{29} \approx 5.3852 \] Thus, the foci are approximately: \[ (5.3852, 0) \text{ and } (-5.3852, 0) \]

Final Answer