The length of the latusrectum of the parbola whose focus is `(3, 3)` and directrix `3x-4y-2=0`, is

Options

A. a) `1`
B. b) `2`
C. c) `4`
D. d) `8`

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 \( (3, 3) \) and the directrix is given by the equation \( 3x - 4y - 2 = 0 \).

### Step 2: Write the formula for the length of the latus rectum
The length of the latus rectum \( l \) of a parabola can be calculated using the formula:
\[ l = 2 \times \text{(perpendicular distance from the focus to the directrix)} \]

### Step 3: Calculate the perpendicular distance from the focus to the directrix
To find the perpendicular distance from the point \( (x_0, y_0) = (3, 3) \) (the focus) to the line \( Ax + By + C = 0 \) (where \( A = 3, B = -4, C = -2 \)), we use the formula:
\[ \text{Distance} = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
Substituting the values:
\[ \text{Distance} = \frac{|3(3) - 4(3) - 2|}{\sqrt{3^2 + (-4)^2}} = \frac{|9 - 12 - 2|}{\sqrt{9 + 16}} = \frac{|-5|}{\sqrt{25}} = \frac{5}{5} = 1 \]

### Step 4: Calculate the length of the latus rectum
Now substituting the distance back into the formula for the latus rectum:
\[ l = 2 \times 1 = 2 \]

### Step 5: Conclusion
Thus, the length of the latus rectum of the parabola is \( 2 \).

### Final Answer
Therefore, the correct option is \( \text{B} \, 2 \).

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