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