The point (3,4) is the focus and 2x-3y+5=0 is the directrix of a parabola . Its latusrectum is

Options

A. `(2)/(sqrt(13))`
B. `(4)/(sqrt(13))`
C. `(1)/(sqrt(13))`
D. `(3)/(sqrt(13))`

Answer

To find the length of the latus rectum of the parabola with focus at (3, 4) and directrix given by the equation \(2x - 3y + 5 = 0\), we can follow these steps:

### Step 1: Identify the focus and directrix
The focus of the parabola is given as \(F(3, 4)\). The directrix is given by the equation \(2x - 3y + 5 = 0\).

### Step 2: Write the equation of the directrix in standard form
To work with the directrix, we can rewrite the equation in the form \(Ax + By + C = 0\):
\[ 2x - 3y + 5 = 0 \]
Here, \(A = 2\), \(B = -3\), and \(C = 5\).

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

### Step 4: Simplify the numerator
Calculating the numerator:
\[ = |6 - 12 + 5| = |6 - 12 + 5| = |-1| = 1 \]

### Step 5: Simplify the denominator
Calculating the denominator:
\[ \sqrt{2^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \]

### Step 6: Calculate the distance
Now substituting back into the distance formula:
\[ d = \frac{1}{\sqrt{13}} \]

### Step 7: Find the length of the latus rectum
The length of the latus rectum \(L\) of the parabola is given by the formula:
\[ L = 2 \times d \]
Substituting the distance we found:
\[ L = 2 \times \frac{1}{\sqrt{13}} = \frac{2}{\sqrt{13}} \]

### Final Answer
Thus, the length of the latus rectum is:
\[ \frac{2}{\sqrt{13}} \]

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