If radius of the director circle of the ellipse `((3x+4y-2)^(2))/(100)+((4x-3y+5)^(2))/(625) =1` is

Options

A. 6
B. `sqrt(34)`
C. `sqrt(29)`
D. `sqrt(26)`

Explanation

To find the radius of the director circle of the given ellipse, we will follow these steps:

### Step 1: Write down the equation of the ellipse
The given equation of the ellipse is:
\[ \frac{(3x + 4y - 2)^2}{100} + \frac{(4x - 3y + 5)^2}{625} = 1 \]

### Step 2: Rewrite the equation in standard form
We can rewrite the equation in a more recognizable form. The denominators can be expressed as squares:
\[ \frac{(3x + 4y - 2)^2}{10^2} + \frac{(4x - 3y + 5)^2}{25^2} = 1 \]
This indicates that \( a = 10 \) and \( b = 25 \).

### Step 3: Identify \( a \) and \( b \)
From the standard form of the ellipse, we can identify:
- \( a = 10 \)
- \( b = 25 \)

### Step 4: Calculate \( a^2 \) and \( b^2 \)
Now we calculate \( a^2 \) and \( b^2 \):
\[ a^2 = 10^2 = 100 \]
\[ b^2 = 25^2 = 625 \]

### Step 5: Use the formula for the radius of the director circle
The radius \( R \) of the director circle for an ellipse is given by the formula:
\[ R = \sqrt{a^2 + b^2} \]

### Step 6: Substitute \( a^2 \) and \( b^2 \) into the formula
Substituting the values we calculated:
\[ R = \sqrt{100 + 625} = \sqrt{725} \]

### Step 7: Simplify \( \sqrt{725} \)
We can simplify \( \sqrt{725} \):
\[ 725 = 25 \times 29 \quad \text{(since } 25 = 5^2\text{)} \]
Thus,
\[ R = \sqrt{25 \times 29} = 5\sqrt{29} \]

### Final Answer
The radius of the director circle of the ellipse is:
\[ R = 5\sqrt{29} \]