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 with focus at (3, 3) and directrix given by the equation \(3x - 4y = 2\), 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 line \(3x - 4y - 2 = 0\).

### Step 2: Use the distance formula
The distance \(d\) from a point \((x_0, y_0)\) to a line \(Ax + By + C = 0\) is given by the formula:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
For our directrix \(3x - 4y - 2 = 0\), we have \(A = 3\), \(B = -4\), and \(C = -2\). We will use the focus point \((3, 3)\) in this formula.

### Step 3: Calculate the distance from the focus to the directrix
Substituting the values into the distance formula:
\[ d = \frac{|3(3) - 4(3) - 2|}{\sqrt{3^2 + (-4)^2}} = \frac{|9 - 12 - 2|}{\sqrt{9 + 16}} = \frac{|-5|}{\sqrt{25}} = \frac{5}{5} = 1 \]
This distance \(d\) represents \(a\), where \(a\) is the distance from the focus to the vertex of the parabola.

### Step 4: Find the length of the latus rectum
The length of the latus rectum \(L\) of a parabola is given by the formula:
\[ L = 4a \]
Since we found \(a = 1\), we can calculate:
\[ L = 4 \times 1 = 4 \]

### Conclusion
Thus, the length of the latus rectum of the parabola is \(4\).

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