Find the center, foci, and vertices of the ellipse. Graph the equation. \( \frac{(x-1)^{2}}{16}+\frac{(y+1)^{2}}{81}=1 \) Type the coordinates of the center of the ellipse in the boxes below. (h,k) \( =\square \) Type the coordinates of the vertices in the boxes below. Vertex above center = \( \square \) (Simplify your answer.) Vertex below center = ( \( \square \) (Simplify your answer.) Type the coordinates of the foci in the boxes below. Focus above center = ( \( \square \) (Simplify your answer.) Focus below center = ( \( \square \). (Simplify your answer.)

To analyze the given ellipse equation: \[ \frac{(x-1)^{2}}{16}+\frac{(y+1)^{2}}{81}=1 \] we can identify the key components of the ellipse. ### Step 1: Identify the center The standard form of an ellipse is: \[ \frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1 \] where \((h, k)\) is the center of the ellipse. From the given equation, we can see that: - \(h = 1\) - \(k = -1\) Thus, the center of the ellipse is: \[ (h, k) = (1, -1) \] ### Step 2: Identify the vertices In the equation, \(a^2 = 16\) and \(b^2 = 81\). Therefore: - \(a = \sqrt{16} = 4\) - \(b = \sqrt{81} = 9\) Since \(b > a\), this indicates that the major axis is vertical. The vertices are located at: - Vertex above center: \((h, k + b) = (1, -1 + 9) = (1, 8)\) - Vertex below center: \((h, k - b) = (1, -1 - 9) = (1, -10)\) ### Step 3: Identify the foci The distance to the foci from the center is given by \(c\), where: \[ c = \sqrt{b^2 - a^2} = \sqrt{81 - 16} = \sqrt{65} \] The foci are located at: - Focus above center: \((h, k + c) = (1, -1 + \sqrt{65})\) - Focus below center: \((h, k - c) = (1, -1 - \sqrt{65})\) ### Summary of Results - Center: \((h, k) = (1, -1)\) - Vertex above center: \((1, 8)\) - Vertex below center: \((1, -10)\) - Focus above center: \((1, -1 + \sqrt{65})\) - Focus below center: \((1, -1 - \sqrt{65})\) ### Final Answers - Center: \( (h, k) = (1, -1) \) - Vertex above center: \( (1, 8) \) - Vertex below center: \( (1, -10) \) - Focus above center: \( (1, -1 + \sqrt{65}) \) - Focus below center: \( (1, -1 - \sqrt{65}) \) ### Graphing the Ellipse To graph the ellipse, plot the center at \((1, -1)\), the vertices at \((1, 8)\) and \((1, -10)\), and the foci at \((1, -1 + \sqrt{65})\) and \((1, -1 - \sqrt{65})\). The ellipse will be elongated vertically, centered at \((1, -1)\).