Find the standard form of the equation of the ellipse satisfying the given conditions. Major axis horizontal with length 12; length of minor axis = 4; center: (-3,5)

Verified step by step guidance

1. Step 1: Recall the standard form of the equation of an ellipse with a horizontal major axis. It is given by: $\backslash frac\{(x-h)^2\}\{a^2\}\; +\; \backslash frac\{(y-k)^2\}\{b^2\}\; =\; 1$, where (h, k) is the center, 'a' is the semi-major axis, and 'b' is the semi-minor axis.
2. Step 2: Identify the center of the ellipse from the problem. The center is given as (-3, 5), so h = -3 and k = 5.
3. Step 3: Determine the values of 'a' and 'b'. The length of the major axis is 12, so the semi-major axis is $a\; =\; \backslash frac\{12\}\{2\}\; =\; 6$. The length of the minor axis is 4, so the semi-minor axis is $b\; =\; \backslash frac\{4\}\{2\}\; =\; 2$.
4. Step 4: Substitute the values of h, k, a, and b into the standard form equation. This gives: $\backslash frac\{(x\; -\; (-3))^2\}\{6^2\}\; +\; \backslash frac\{(y\; -\; 5)^2\}\{2^2\}\; =\; 1$.
5. Step 5: Simplify the equation. Replace $6^2$ with 36 and $2^2$ with 4, and simplify $x\; -\; (-3)$ to $x\; +\; 3$. The equation becomes: $\backslash frac\{(x\; +\; 3)^2\}\{36\}\; +\; \backslash frac\{(y\; -\; 5)^2\}\{4\}\; =\; 1$.

Key Concepts

Standard Form of an Ellipse

The standard form of the equation of an ellipse is given by the formula \\( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \\ where (h, k) is the center of the ellipse, 'a' is half the length of the major axis, and 'b' is half the length of the minor axis. This form allows for easy identification of the ellipse's dimensions and orientation.

Major and Minor Axes

The major axis of an ellipse is the longest diameter, while the minor axis is the shortest. In this case, the major axis is horizontal with a length of 12, meaning 'a' equals 6 (half of 12), and the minor axis has a length of 4, making 'b' equal 2 (half of 4). The orientation of these axes determines the structure of the ellipse's equation.

Center of the Ellipse

The center of the ellipse is the midpoint of both axes and is represented by the coordinates (h, k) in the standard form equation. For this problem, the center is given as (-3, 5), which means that in the standard form, 'h' is -3 and 'k' is 5. This point is crucial for accurately positioning the ellipse on the coordinate plane.