The length of major ofthe ellipse `(5x-10)^2 +(5y+15)^2 = 1/4(3x-4y+7)^2` is

Options

A. 10
B. `20//3`
C. `20//7`
D. 4

Explanation

To find the length of the major axis of the ellipse given by the equation

\[ (5x - 10)^2 + (5y + 15)^2 = \frac{1}{4}(3x - 4y + 7)^2, \]

we will follow these steps:

### Step 1: Rewrite the equation
Start by rewriting the equation in a more manageable form. We can divide both sides by 4 to simplify:

\[ \frac{(5x - 10)^2}{4} + \frac{(5y + 15)^2}{4} = \frac{(3x - 4y + 7)^2}{16}. \]

### Step 2: Simplify the left side
Next, we can simplify the left side of the equation:

\[ \left(\frac{5x - 10}{2}\right)^2 + \left(\frac{5y + 15}{2}\right)^2 = \frac{(3x - 4y + 7)^2}{16}. \]

Let \( X = \frac{5x - 10}{2} \) and \( Y = \frac{5y + 15}{2} \). Then we have:

\[ X^2 + Y^2 = \frac{(3x - 4y + 7)^2}{16}. \]

### Step 3: Identify the center and axes
From the transformation, we can find the center of the ellipse. The center is located at:

\[ \left(2, -3\right). \]

### Step 4: Find the directrix
The directrix can be derived from the equation \( 3x - 4y + 7 = 0 \). We need to find the distance from the focus to the directrix.

### Step 5: Calculate the distance from the focus to the directrix
The distance \( d \) from the focus \( (2, -3) \) to the directrix can be calculated using the formula for the distance from a point to a line:

\[ d = \frac{|3(2) - 4(-3) + 7|}{\sqrt{3^2 + (-4)^2}} = \frac{|6 + 12 + 7|}{\sqrt{9 + 16}} = \frac{25}{5} = 5. \]

### Step 6: Relate distance to semi-major axis
The relationship between the semi-major axis \( a \) and the distance to the directrix \( d \) is given by:

\[ d = \frac{a}{e} \quad \text{and} \quad e = \frac{c}{a}, \]

where \( c \) is the distance from the center to the focus.

### Step 7: Solve for \( a \)
Using the relationship, we find:

\[ 5 = \frac{a}{e} \quad \Rightarrow \quad a = 5e. \]

### Step 8: Find \( c \) and \( e \)
From the ellipse properties, we know:

\[ c = \sqrt{a^2 - b^2}. \]

Using the relationship \( e = \frac{c}{a} \), we can relate \( a \) and \( c \) to find the length of the major axis.

### Step 9: Calculate the length of the major axis
The length of the major axis is given by:

\[ \text{Length of major axis} = 2a. \]

From the previous calculations, we find:

\[ a = \frac{10}{3} \quad \Rightarrow \quad \text{Length of major axis} = 2 \times \frac{10}{3} = \frac{20}{3}. \]

### Final Answer
Thus, the length of the major axis of the ellipse is

\[ \frac{20}{3}. \]