To determine the vertices of the ellipse given by the equation [tex]\(\frac{(x-7)^2}{64} + \frac{(y+2)^2}{9} = 1\)[/tex], we need to analyze the standard form of the ellipse equation and identify the necessary parameters.
The standard form of an ellipse's equation is:
[tex]\[
\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1
\][/tex]
where [tex]\((h, k)\)[/tex] represents the center of the ellipse, [tex]\(a\)[/tex] is the length of the semi-major axis, and [tex]\(b\)[/tex] is the length of the semi-minor axis. Here's the detailed step-by-step breakdown:
1. Identify the center of the ellipse:
The given equation is [tex]\(\frac{(x-7)^2}{64} + \frac{(y+2)^2}{9} = 1\)[/tex].
Here, [tex]\(h = 7\)[/tex] and [tex]\(k = -2\)[/tex]. Therefore, the center of the ellipse is [tex]\((7, -2)\)[/tex].
2. Determine the lengths of the semi-major axis and the semi-minor axis:
In the equation [tex]\(\frac{(x-7)^2}{64} + \frac{(y+2)^2}{9} = 1\)[/tex],
[tex]\[
a^2 = 64 \quad \Rightarrow \quad a = \sqrt{64} = 8
\][/tex]
[tex]\[
b^2 = 9 \quad \Rightarrow \quad b = \sqrt{9} = 3
\][/tex]
3. Identify the orientation of the ellipse:
Since [tex]\(a > b\)[/tex], the major axis of the ellipse lies along the x-axis.
4. Find the vertices along the major axis:
Since the major axis is horizontal, the vertices are horizontally aligned with the center.
The vertices are located [tex]\(a\)[/tex] units to the left and right of the center along the x-axis.
Therefore, we calculate the vertices as follows:
[tex]\[
(h - a, k) \quad \text{and} \quad (h + a, k)
\][/tex]
Substituting [tex]\(h = 7\)[/tex], [tex]\(k = -2\)[/tex], and [tex]\(a = 8\)[/tex]:
[tex]\[
(7 - 8, -2) = (-1, -2)
\][/tex]
[tex]\[
(7 + 8, -2) = (15, -2)
\][/tex]
Therefore, the vertices of the ellipse are [tex]\((-1, -2)\)[/tex] and [tex]\((15, -2)\)[/tex]. The correct answer is:
[tex]\[
(15, -2) \quad \text{and} \quad (-1, -2)
\][/tex]
