The parabola `y^(2) = 2ax ` passes through the point ` (-2,1)` .The length of its latus rectum is

Options

A. A. 2 units
B. B. `(1)/(2) ` units
C. C. 4 units
D. D. `(1)/(4)` units

Answer

To solve the problem step by step, we will follow the reasoning outlined in the video transcript.

### Step 1: Understand the given parabola
The equation of the parabola is given as:
\[ y^2 = 2ax \]
This is a standard form of a parabola that opens to the right.

### Step 2: Substitute the point into the parabola's equation
We know that the parabola passes through the point \((-2, 1)\). We will substitute \(x = -2\) and \(y = 1\) into the parabola's equation:
\[ 1^2 = 2a(-2) \]

### Step 3: Simplify the equation
Now, simplify the equation:
\[ 1 = 2a(-2) \]
\[ 1 = -4a \]

### Step 4: Solve for \(a\)
To find the value of \(a\), we rearrange the equation:
\[ a = -\frac{1}{4} \]

### Step 5: Find the length of the latus rectum
The length of the latus rectum \(L\) of a parabola given by the equation \(y^2 = 2ax\) is given by:
\[ L = 4|a| \]
Substituting the value of \(a\):
\[ L = 4 \left| -\frac{1}{4} \right| \]
\[ L = 4 \times \frac{1}{4} \]
\[ L = 1 \]

### Final Answer
The length of the latus rectum is:
\[ \boxed{1} \]