To find the equation of the ellipse with the given center, focus, and vertex, we need to determine the values for [tex]\(A\)[/tex], [tex]\(B\)[/tex], [tex]\(C\)[/tex], and [tex]\(D\)[/tex] and plug them into the standard form equation of an ellipse.
The standard form equation of an ellipse with a horizontal major axis centered at [tex]\((C, D)\)[/tex] is:
[tex]\[ \frac{(x - C)^2}{A^2} + \frac{(y - D)^2}{B^2} = 1 \][/tex]
Given:
- The center of the ellipse is [tex]\((C, D) = (1, 5)\)[/tex]. So, [tex]\(C = 1\)[/tex] and [tex]\(D = 5\)[/tex].
- The vertex [tex]\((6, 5)\)[/tex] lies on the horizontal axis. The distance from the center [tex]\((1,5)\)[/tex] to the vertex [tex]\((6,5)\)[/tex] is 5 units. Therefore, [tex]\(A = 5\)[/tex].
- The focus is at [tex]\((-2, 5)\)[/tex]. The distance from the center [tex]\((1, 5)\)[/tex] to the focus [tex]\((-2, 5)\)[/tex] is 3 units. This distance is denoted by [tex]\(c\)[/tex], so [tex]\( c = 3 \)[/tex].
The relationship between [tex]\(A\)[/tex], [tex]\(B\)[/tex], and the distance to the focus, [tex]\(c\)[/tex], is given by:
[tex]\[ c^2 = A^2 - B^2 \][/tex]
We know [tex]\(A = 5\)[/tex] and [tex]\(c = 3\)[/tex], so we substitute these values into the equation:
[tex]\[ 3^2 = 5^2 - B^2 \][/tex]
[tex]\[ 9 = 25 - B^2 \][/tex]
[tex]\[ B^2 = 25 - 9 \][/tex]
[tex]\[ B^2 = 16 \][/tex]
[tex]\[ B = 4 \][/tex]
To summarize, the values are:
[tex]\[ A = 5 \][/tex]
[tex]\[ B = 4 \][/tex]
[tex]\[ C = 1 \][/tex]
[tex]\[ D = 5 \][/tex]
Now, substituting these values into the standard ellipse equation:
[tex]\[ \frac{(x - C)^2}{A^2} + \frac{(y - D)^2}{B^2} = 1 \][/tex]
[tex]\[ \frac{(x - 1)^2}{5^2} + \frac{(y - 5)^2}{4^2} = 1 \][/tex]
[tex]\[ \frac{(x - 1)^2}{25} + \frac{(y - 5)^2}{16} = 1 \][/tex]
Thus, the equation of the ellipse is:
[tex]\[ \frac{(x - 1)^2}{25} + \frac{(y - 5)^2}{16} = 1 \][/tex]
The values for [tex]\(A\)[/tex], [tex]\(B\)[/tex], [tex]\(C\)[/tex], and [tex]\(D\)[/tex] are:
[tex]\[ A = 5 \][/tex]
[tex]\[ B = 4 \][/tex]
[tex]\[ C = 1 \][/tex]
[tex]\[ D = 5 \][/tex]
