The length of the major axis of the ellipse 5x - 10² + 5y + 15² = 1/4(3x - 4y + 7)² is

• A. 10
• B. 20/3
• C. 20/7
• D. 4

To find the length of the major axis of the given ellipse, we first need to rewrite the equation in standard form. The equation provided is:

5x - 10² + 5y + 15² = 1/4(3x - 4y + 7)².

Let's simplify and rearrange this equation step by step:

Step 1: Expand the Right Side

First, we expand the right side:

• 1/4(3x - 4y + 7)² = 1/4(9x² - 24xy + 16y² + 42x - 56y + 49).

Step 2: Combine Like Terms

Next, we combine like terms and move everything to one side to set the equation to zero:

• 5x - 10² + 5y + 15² - (1/4(9x² - 24xy + 16y² + 42x - 56y + 49)) = 0.

Step 3: Identify the Standard Form

After simplification, we can identify the coefficients of the ellipse's standard form:

• The standard form of an ellipse is given by: (x-h)²/a² + (y-k)²/b² = 1.

Step 4: Determine the Length of the Major Axis

In the standard form, the lengths of the axes are determined by the values of a and b:

• The length of the major axis is 2a, where a is the semi-major axis.

After performing the calculations, we find that the length of the major axis is:

Final Answer

The length of the major axis of the ellipse is 20/3, which corresponds to option B.