If the parabola `y^(2)=4ax` passes through the point (2,-3) then find the co-ordinates of the focus and the length of latus rectum.

Answer

To solve the problem step by step, we will follow these instructions:

### Step 1: Substitute the point into the parabola equation
The given parabola is \( y^2 = 4ax \). We need to check if the point (2, -3) lies on this parabola.

Substituting \( x = 2 \) and \( y = -3 \) into the equation:
\[ (-3)^2 = 4a(2) \]
This simplifies to:
\[ 9 = 8a \]

### Step 2: Solve for \( a \)
From the equation \( 9 = 8a \), we can solve for \( a \):
\[ a = \frac{9}{8} \]

### Step 3: Write the equation of the parabola
Now that we have the value of \( a \), we can write the equation of the parabola:
\[ y^2 = 4 \left(\frac{9}{8}\right)x \]
This simplifies to:
\[ y^2 = \frac{36}{8}x \quad \text{or} \quad y^2 = \frac{9}{2}x \]

### Step 4: Find the coordinates of the focus
For the parabola \( y^2 = 4ax \), the coordinates of the focus are given by \( (a, 0) \). Thus:
\[ \text{Focus} = \left(\frac{9}{8}, 0\right) \]

### Step 5: Find the length of the latus rectum
The length of the latus rectum \( L \) for a parabola is given by \( 4a \). Therefore:
\[ L = 4 \left(\frac{9}{8}\right) = \frac{36}{8} = \frac{9}{2} \]

### Final Answers
- The coordinates of the focus are \( \left(\frac{9}{8}, 0\right) \).
- The length of the latus rectum is \( \frac{9}{2} \).

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