If the parabola `y^(2)=4ax` passes through the point `(4,5)`, find the length of its latus rectum.

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the equation of the parabola
The equation of the parabola is given as \( y^2 = 4ax \). Here, \( a \) is a constant that determines the width of the parabola.

### Step 2: Substitute the point into the parabola's equation
We know that the parabola passes through the point \( (4, 5) \). To find the value of \( a \), we will substitute \( x = 4 \) and \( y = 5 \) into the equation of the parabola.

\[ y^2 = 4ax \]
Substituting the point:
\[ 5^2 = 4a \cdot 4 \]

### Step 3: Calculate \( y^2 \)
Calculating \( 5^2 \):
\[ 25 = 4a \cdot 4 \]

### Step 4: Simplify the equation
Now, simplify the equation:
\[ 25 = 16a \]

### Step 5: Solve for \( a \)
To find \( a \), divide both sides by 16:
\[ a = \frac{25}{16} \]

### Step 6: Find the length of the latus rectum
The length of the latus rectum of a parabola \( y^2 = 4ax \) is given by the formula \( 4a \). Now that we have found \( a \), we can calculate \( 4a \):
\[ 4a = 4 \cdot \frac{25}{16} = \frac{100}{16} = \frac{25}{4} \]

### Final Answer
The length of the latus rectum is \( \frac{25}{4} \).

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