The length of the latus rectum of the parabola whose focus is `(3,3)` and directrix is `3x-4y-2=0` is.

Options

A. `2`
B. `1`
C. `4`
D. None of these

Answer

To find the length of the latus rectum of the parabola with the given focus and directrix, we can follow these steps:

### Step 1: Identify the focus and directrix
The focus of the parabola is given as \( F(3, 3) \) and the directrix is given by the equation \( 3x - 4y - 2 = 0 \).

### Step 2: Use the distance formula
The distance \( d \) between a point \( (x_1, y_1) \) and a line \( ax + by + c = 0 \) is given by the formula:
\[ d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} \]
In our case, \( (x_1, y_1) = (3, 3) \), \( a = 3 \), \( b = -4 \), and \( c = -2 \).

### Step 3: Substitute the values into the formula
Substituting the values into the distance formula:
\[ d = \frac{|3 \cdot 3 + (-4) \cdot 3 - 2|}{\sqrt{3^2 + (-4)^2}} \]

### Step 4: Calculate the numerator
Calculating the numerator:
\[ 3 \cdot 3 = 9, \quad -4 \cdot 3 = -12 \quad \Rightarrow \quad 9 - 12 - 2 = -5 \]
So, the absolute value is:
\[ | -5 | = 5 \]

### Step 5: Calculate the denominator
Calculating the denominator:
\[ \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 6: Calculate the distance \( d \)
Now substituting back into the distance formula:
\[ d = \frac{5}{5} = 1 \]

### Step 7: Relate distance to \( a \)
The distance \( d \) between the focus and the directrix is equal to \( 2a \):
\[ 2a = 1 \quad \Rightarrow \quad a = \frac{1}{2} \]

### Step 8: Calculate the length of the latus rectum
The length of the latus rectum \( L \) is given by the formula:
\[ L = 4a \]
Substituting the value of \( a \):
\[ L = 4 \cdot \frac{1}{2} = 2 \]

### Final Answer
The length of the latus rectum of the parabola is \( 2 \).

---