Find the standard form of the equation of the ellipse satisfying the given conditions.
Foci: $(0,-7),(0,7)$; vertices: $(0,-9),(0,9)$
Answer
Final Answer: The standard form of the equation of the ellipse is \(\boxed{\frac{x^2}{31.999999999999996} + \frac{y^2}{81} = 1}\).
Step 1 ：The standard form of the equation of an ellipse with its center at the origin (0,0) is given by \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). The value of \(a\) is the distance from the center to a vertex, and the value of \(b\) is the distance from the center to a co-vertex. The value of \(c\) is the distance from the center to a focus. For an ellipse, \(a^2 = b^2 + c^2\).
Step 2 ：Given that the foci are at \((0,-7),(0,7)\), we can determine that \(c = 7\). The vertices are at \((0,-9),(0,9)\), so \(a = 9\). We can use these values to find \(b\) using the equation \(a^2 = b^2 + c^2\).
Step 3 ：Substituting \(a = 9\) and \(c = 7\) into the equation, we get \(b = \sqrt{a^2 - c^2} = \sqrt{9^2 - 7^2} = \sqrt{81 - 49} = \sqrt{32} = 5.656854249492381\).
Step 4 ：Now that we have the values for \(a\), \(b\), and \(c\), we can substitute them into the standard form of the equation for an ellipse. Since the foci and vertices are aligned along the y-axis, the equation will be \(\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\).
Step 5 ：Substituting \(a = 9\), \(b = 5.656854249492381\) into the equation, we get \(\frac{x^2}{(5.656854249492381)^2} + \frac{y^2}{9^2} = 1\), which simplifies to \(\frac{x^2}{31.999999999999996} + \frac{y^2}{81} = 1\).
Step 6 ：Final Answer: The standard form of the equation of the ellipse is \(\boxed{\frac{x^2}{31.999999999999996} + \frac{y^2}{81} = 1}\).
