The focus of the parabola `2y^(2) = 5x ` is ............ .

Answer

To find the focus of the parabola given by the equation \(2y^2 = 5x\), we will follow these steps:

### Step 1: Rewrite the equation in standard form
We start with the equation of the parabola:
\[ 2y^2 = 5x \]
To put it in standard form, we divide both sides by 5:
\[ \frac{2y^2}{5} = x \]
This can be rewritten as:
\[ y^2 = \frac{5}{2}x \]

### Step 2: Identify the parameters of the parabola
The standard form of a parabola that opens to the right is given by:
\[ y^2 = 4ax \]
From our equation \(y^2 = \frac{5}{2}x\), we can identify \(4a = \frac{5}{2}\).

### Step 3: Solve for \(a\)
To find \(a\), we solve the equation:
\[ 4a = \frac{5}{2} \]
Dividing both sides by 4 gives:
\[ a = \frac{5}{2} \cdot \frac{1}{4} = \frac{5}{8} \]

### Step 4: Determine the focus
For a parabola in the form \(y^2 = 4ax\), the focus is located at the point \((a, 0)\). Therefore, substituting our value of \(a\):
\[ \text{Focus} = \left(\frac{5}{8}, 0\right) \]

### Final Answer
The focus of the parabola \(2y^2 = 5x\) is \(\left(\frac{5}{8}, 0\right)\).

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