The latusrectum of the parabola `13[(x-3)^(2)+(y-4)^(2) ]= (2x-3y+5)^(2)`

Options

A. `(2)/(sqrt(13))`
B. `(4)/(sqrt(13))`
C. `(1)/(sqrt(13))`
D. `(5)/(sqrt(13))`

Answer

To find the length of the latus rectum of the given parabola, we will follow these steps:

### Step 1: Write the equation of the parabola
The given equation of the parabola is:
\[ 13[(x-3)^{2} + (y-4)^{2}] = (2x - 3y + 5)^{2} \]

### Step 2: Rearrange the equation
We can rearrange the equation by taking the square root of both sides:
\[ \sqrt{13} \sqrt{(x-3)^{2} + (y-4)^{2}} = |2x - 3y + 5| \]

### Step 3: Identify the points
Let \( P(x, y) \) be a point on the parabola and \( S(3, 4) \) be the focus of the parabola. The expression on the left side represents the distance from point \( P \) to point \( S \).

### Step 4: Use the distance formula
The length of the latus rectum is given by the formula:
\[ \text{Length of latus rectum} = 2 \times \text{perpendicular distance from focus to directrix} \]

### Step 5: Calculate the perpendicular distance
The perpendicular distance from point \( S(3, 4) \) to the line \( 2x - 3y + 5 = 0 \) can be calculated using the formula:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
where \( A = 2, B = -3, C = 5 \), and \( (x_1, y_1) = (3, 4) \).

### Step 6: Substitute values into the distance formula
Substituting the values into the distance formula:
\[ d = \frac{|2(3) - 3(4) + 5|}{\sqrt{2^2 + (-3)^2}} \]
\[ = \frac{|6 - 12 + 5|}{\sqrt{4 + 9}} \]
\[ = \frac{| -1 |}{\sqrt{13}} \]
\[ = \frac{1}{\sqrt{13}} \]

### Step 7: Calculate the length of the latus rectum
Now, we can find the length of the latus rectum:
\[ \text{Length of latus rectum} = 2 \times d = 2 \times \frac{1}{\sqrt{13}} = \frac{2}{\sqrt{13}} \]

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

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