Questions: What are the vertices of the hyperbola defined by the equation x^2/4 - y^2/25 = 1 ? (0,0) (2,0) and (-2,0) (0,5) and (0,-5) (2,5) and (-2,-5) (4,25) and (-4,-25)

Solution Steps

The given equation is \(\frac{x^{2}}{4} - \frac{y^{2}}{25} = 1\). This is in the standard form of a hyperbola \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\), where the transverse axis is horizontal.

From the equation \(\frac{x^{2}}{4} - \frac{y^{2}}{25} = 1\), we identify \(a^{2} = 4\) and \(b^{2} = 25\). Therefore, \(a = 2\) and \(b = 5\).

For a hyperbola of the form \(\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\), the vertices are located at \((\pm a, 0)\). Substituting the value of \(a\), the vertices are \((\pm 2, 0)\), which corresponds to the points \((2, 0)\) and \((-2, 0)\).

Final Answer