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 can follow these steps:

### Step 1: Identify 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:
\[ \frac{(3x + 4y - 2)^2}{10^2} + \frac{(4x - 3y + 5)^2}{25^2} = 1 \]

### Step 3: Compare with the standard form of the ellipse
The standard form of an ellipse is given by:
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
From our equation, we can identify \(a^2\) and \(b^2\):
- \(a^2 = 100\) (which corresponds to the first term)
- \(b^2 = 625\) (which corresponds to the second term)

### Step 4: Calculate \(a\) and \(b\)
Taking the square roots:
- \(a = \sqrt{100} = 10\)
- \(b = \sqrt{625} = 25\)

### Step 5: Find the radius of the director circle
The radius \(R\) of the director circle of an ellipse is given by the formula:
\[ R = \sqrt{a^2 + b^2} \]
Substituting the values we found:
\[ R = \sqrt{100 + 625} = \sqrt{725} \]

### Step 6: Simplify \(\sqrt{725}\)
We can simplify \(\sqrt{725}\):
\[ \sqrt{725} = \sqrt{25 \times 29} = 5\sqrt{29} \]

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