Radius of the circle `9x^(2)+y^(2)=4(x^(2)-y^(2))-8x` is

Options

A. `(2)/(5)`
B. `(4)/(5)`
C. `(8)/(5)`
D. none

Answer

To find the radius of the circle given by the equation \( 9x^2 + y^2 = 4(x^2 - y^2) - 8x \), we will first rearrange the equation into the standard form of a circle.

### Step 1: Rearrange the equation
Start with the given equation:
\[ 9x^2 + y^2 = 4(x^2 - y^2) - 8x \]
Distributing on the right side:
\[ 9x^2 + y^2 = 4x^2 - 4y^2 - 8x \]

### Step 2: Move all terms to one side
Rearranging gives:
\[ 9x^2 + y^2 - 4x^2 + 4y^2 + 8x = 0 \]
Combine like terms:
\[ (9x^2 - 4x^2) + (y^2 + 4y^2) + 8x = 0 \]
This simplifies to:
\[ 5x^2 + 5y^2 + 8x = 0 \]

### Step 3: Divide by 5
To simplify, divide the entire equation by 5:
\[ x^2 + y^2 + \frac{8}{5}x = 0 \]

### Step 4: Complete the square for the x terms
To complete the square for the \(x\) terms, we rewrite:
\[ x^2 + \frac{8}{5}x + y^2 = 0 \]
The term to complete the square for \(x^2 + \frac{8}{5}x\) is \(\left(\frac{4}{5}\right)^2 = \frac{16}{25}\):
\[ \left(x + \frac{4}{5}\right)^2 - \frac{16}{25} + y^2 = 0 \]
Rearranging gives:
\[ \left(x + \frac{4}{5}\right)^2 + y^2 = \frac{16}{25} \]

### Step 5: Identify the center and radius
Now, we have the equation in the standard form of a circle:
\[ \left(x - h\right)^2 + \left(y - k\right)^2 = r^2 \]
where \((h, k)\) is the center and \(r\) is the radius. From our equation:
- Center: \(\left(-\frac{4}{5}, 0\right)\)
- Radius: \(r = \sqrt{\frac{16}{25}} = \frac{4}{5}\)

### Final Answer
The radius of the circle is:
\[ \frac{4}{5} \]

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