Question

question 7 of 10 which of the following equations represents an ellipse having vertices located at (2,9) and (2,-5) and foci located at (2,5) and (2,-1)? a. $\frac{(x + 2)^2}{40}+\frac{(y + 2)^2}{49}=1$ b. $\frac{(x - 2)^2}{49}+\frac{(y - 2)^2}{40}=1$ c. $\frac{(x - 2)^2}{40}+\frac{(y - 2)^2}{49}=1$ d. $\frac{(x - 2)^2}{9}+\frac{(y - 2)^2}{49}=1$

Explanation:

Step1: Determine the center of the ellipse

The center of the ellipse is the mid - point between the vertices or foci. For points \((x_1,y_1)\) and \((x_2,y_2)\), the mid - point formula is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). For vertices \((2,9)\) and \((2,-5)\), the x - coordinate of the center \(h=\frac{2 + 2}{2}=2\), and the y - coordinate of the center \(k=\frac{9+( - 5)}{2}=\frac{4}{2}=2\). So the center of the ellipse is \((2,2)\).

Step2: Find the value of \(a\)

The distance between the center \((2,2)\) and a vertex \((2,9)\) (or \((2,-5)\)) gives the value of \(a\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), here \(x_1 = 2,y_1 = 2,x_2 = 2,y_2 = 9\), so \(a=\vert9 - 2\vert=7\), and \(a^{2}=49\).

Step3: Find the value of \(c\)

The distance between the center \((2,2)\) and a focus \((2,5)\) (or \((2,-1)\)) gives the value of \(c\). Using the distance formula, with \(x_1 = 2,y_1 = 2,x_2 = 2,y_2 = 5\), \(c=\vert5 - 2\vert=3\).

Step4: Find the value of \(b^{2}\)

We know the relationship \(c^{2}=a^{2}-b^{2}\) for an ellipse. Substituting \(a = 7\) and \(c = 3\) into the formula \(3^{2}=7^{2}-b^{2}\), we get \(b^{2}=a^{2}-c^{2}=49 - 9=40\).

Step5: Write the equation of the ellipse

The standard form of the equation of an ellipse with a vertical major axis (since the vertices and foci have the same x - coordinate) is \(\frac{(x - h)^{2}}{b^{2}}+\frac{(y - k)^{2}}{a^{2}}=1\), where \((h,k)\) is the center of the ellipse. Substituting \(h = 2,k = 2,a^{2}=49,b^{2}=40\) into the formula, we get \(\frac{(x - 2)^{2}}{40}+\frac{(y - 2)^{2}}{49}=1\).

Answer:

C. \(\frac{(x - 2)^{2}}{40}+\frac{(y - 2)^{2}}{49}=1\)