If the focus of a parabola is `(3,3)` and its directrix is `3x-4y=2` then the length of its latus rectum is

Options

A. 2
B. 4
C. 3
D. 5

Explanation

To find the length of the latus rectum of the parabola given its 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 \( 3x - 4y = 2 \).

### Step 2: Convert the directrix equation to standard form
We can rewrite the directrix equation in the form \( Ax + By + C = 0 \):
\[ 3x - 4y - 2 = 0 \]
Here, \( A = 3 \), \( B = -4 \), and \( C = -2 \).

### Step 3: Calculate the perpendicular distance from the focus to the directrix
The formula for the perpendicular distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
Substituting \( (x_0, y_0) = (3, 3) \):
\[ d = \frac{|3(3) - 4(3) - 2|}{\sqrt{3^2 + (-4)^2}} \]
Calculating the numerator:
\[ = \frac{|9 - 12 - 2|}{\sqrt{9 + 16}} = \frac{|-5|}{\sqrt{25}} = \frac{5}{5} = 1 \]

### Step 4: Relate the distance to \( 2a \)
The distance \( d \) we calculated is equal to \( 2a \) where \( a \) is the distance from the vertex to the focus. Thus:
\[ 2a = 1 \implies a = \frac{1}{2} \]

### Step 5: Calculate the length of the latus rectum
The length of the latus rectum \( L \) of a parabola is given by:
\[ L = 4a \]
Substituting the value of \( a \):
\[ L = 4 \times \frac{1}{2} = 2 \]

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

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