In the xy-plane, the graph of `2x^2 - 6x + 2y^2 =45` is a circle . What is the radius of the circle ?

Options

A. 5
B. 6.5
C. `sqrt40`
D. `sqrt50`

Explanation

To find the radius of the circle represented by the equation \(2x^2 - 6x + 2y^2 = 45\), we need to rewrite this equation in the standard form of a circle. The standard form of a circle is given by:

\[ (x - a)^2 + (y - b)^2 = r^2 \]

where \((a, b)\) is the center of the circle and \(r\) is the radius.

### Step 1: Rewrite the equation
Start with the given equation:

\[ 2x^2 - 6x + 2y^2 = 45 \]

We can factor out the 2 from the left side:

\[ 2(x^2 - 3x + y^2) = 45 \]

Now, divide both sides by 2:

\[ x^2 - 3x + y^2 = \frac{45}{2} \]

### Step 2: Complete the square for \(x\)
To complete the square for the \(x\) terms, we take the coefficient of \(x\) (which is -3), halve it, and square it:

\[ \left(-\frac{3}{2}\right)^2 = \frac{9}{4} \]

Add and subtract \(\frac{9}{4}\) inside the equation:

\[ x^2 - 3x + \frac{9}{4} + y^2 = \frac{45}{2} + \frac{9}{4} \]

### Step 3: Simplify the equation
Now, rewrite the left side as a complete square:

\[ \left(x - \frac{3}{2}\right)^2 + y^2 = \frac{45}{2} + \frac{9}{4} \]

To add the fractions on the right side, convert \(\frac{45}{2}\) to have a common denominator of 4:

\[ \frac{45}{2} = \frac{90}{4} \]

Now add:

\[ \frac{90}{4} + \frac{9}{4} = \frac{99}{4} \]

So the equation now looks like:

\[ \left(x - \frac{3}{2}\right)^2 + y^2 = \frac{99}{4} \]

### Step 4: Identify \(r^2\) and find \(r\)
From the equation, we see that:

\[ r^2 = \frac{99}{4} \]

To find \(r\), take the square root:

\[ r = \sqrt{\frac{99}{4}} = \frac{\sqrt{99}}{2} \]

### Step 5: Simplify \(\sqrt{99}\)
We can simplify \(\sqrt{99}\):

\[ \sqrt{99} = \sqrt{9 \times 11} = 3\sqrt{11} \]

Thus, the radius \(r\) becomes:

\[ r = \frac{3\sqrt{11}}{2} \]

### Final Answer
The radius of the circle is:

\[ \frac{3\sqrt{11}}{2} \]