Length of latus rectum of a parabola whose focus is (3,3) and directrix is ` 3x - 4y - 2 = 0 ` is

Options

A. 1
B. 2
C. 3
D. 4

Answer

To find the length of the latus rectum of the parabola whose focus is at (3, 3) and directrix is given by the equation \(3x - 4y - 2 = 0\), we can follow these steps:

### Step 1: Identify the coefficients of the directrix
The directrix is given in the form \(Ax + By + C = 0\), where:
- \(A = 3\)
- \(B = -4\)
- \(C = -2\)

### Step 2: Calculate the distance from the focus to the directrix
The distance \(d\) from the point (focus) \((x_1, y_1) = (3, 3)\) to the directrix can be calculated using the formula:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
Substituting the values:
\[ d = \frac{|3 \cdot 3 + (-4) \cdot 3 - 2|}{\sqrt{3^2 + (-4)^2}} \]
Calculating the numerator:
\[ = |9 - 12 - 2| = |-5| = 5 \]
Calculating the denominator:
\[ = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Thus, the distance \(d\) is:
\[ d = \frac{5}{5} = 1 \]

### Step 3: Relate the distance to the parameter \(a\)
In a parabola, the distance from the focus to the directrix is equal to \(2a\). Therefore, we have:
\[ 2a = 1 \implies a = \frac{1}{2} \]

### Step 4: Calculate the length of the latus rectum
The length of the latus rectum \(L\) of a parabola 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 \(2\).

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