The equation of directrix and latusrectum of a parabola are 3x - 4y + 27 = 0 and 3x - 4y + 2 = 0 then length of latusrectum is

Options

A. 5
B. 10
C. 15
D. 20

Answer

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

### Step-by-Step Solution:

1. **Identify the equations**:
- The equation of the directrix is given as \(3x - 4y + 27 = 0\).
- The equation of the latus rectum is given as \(3x - 4y + 2 = 0\).

2. **Identify the constants**:
- From the directrix equation, we can identify \(c_1 = 27\).
- From the latus rectum equation, we can identify \(c_2 = 2\).

3. **Use the distance formula**:
The distance \(d\) between two parallel lines of the form \(Ax + By + C_1 = 0\) and \(Ax + By + C_2 = 0\) is given by:
\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]
Here, \(A = 3\), \(B = -4\), \(C_1 = 27\), and \(C_2 = 2\).

4. **Calculate the distance**:
- Calculate the absolute difference:
\[ |C_2 - C_1| = |2 - 27| = |-25| = 25 \]
- Calculate the denominator:
\[ \sqrt{A^2 + B^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
- Now, substitute these values into the distance formula:
\[ d = \frac{25}{5} = 5 \]

5. **Calculate the length of the latus rectum**:
The length of the latus rectum \(L\) is given by:
\[ L = 2d \]
Therefore:
\[ L = 2 \times 5 = 10 \]

### Final Answer:
The length of the latus rectum is \(10\).