Two tangents on a parabola are x-y=0 and x+y=0.
S(2,3) is the focus of the parabola.
The length of latus rectum of the parabola is

Options

A. (a) `6//sqrt(3)`
B. (b) `10//sqrt(13)`
C. (c) `2//sqrt(13)`
D. (d) none of these

Explanation

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

### Step 1: Identify the tangents and their slopes
The equations of the tangents are:
1. \( x - y = 0 \) (which can be rewritten as \( y = x \))
2. \( x + y = 0 \) (which can be rewritten as \( y = -x \))

The slopes of these tangents are:
- Slope of the first tangent \( m_1 = 1 \)
- Slope of the second tangent \( m_2 = -1 \)

### Step 2: Find the coordinates of the foot of the perpendicular from the focus to the tangents
The focus of the parabola is given as \( S(2, 3) \).

#### For the first tangent \( x - y = 0 \):
Using the formula for the foot of the perpendicular from a point \( (x_0, y_0) \) to a line \( Ax + By + C = 0 \):
\[ \left( \frac{B(Bx_0 - Ay_0) - AC}{A^2 + B^2}, \frac{A(Ay_0 - Bx_0) - BC}{A^2 + B^2} \right) \]
For the line \( x - y = 0 \) (or \( 1x + (-1)y + 0 = 0 \)):
- \( A = 1, B = -1, C = 0 \)
- \( (x_0, y_0) = (2, 3) \)

Calculating the foot of the perpendicular:
\[ x_1 = \frac{-1(-1 \cdot 2 - 1 \cdot 3) - 0 \cdot 1}{1^2 + (-1)^2} = \frac{5}{2} \]
\[ y_1 = \frac{1(1 \cdot 3 - (-1) \cdot 2) - 0 \cdot (-1)}{1^2 + (-1)^2} = \frac{5}{2} \]
Thus, the foot of the perpendicular from the focus to the first tangent is \( \left( \frac{5}{2}, \frac{5}{2} \right) \).

#### For the second tangent \( x + y = 0 \):
Using the same formula:
For the line \( x + y = 0 \) (or \( 1x + 1y + 0 = 0 \)):
- \( A = 1, B = 1, C = 0 \)

Calculating the foot of the perpendicular:
\[ x_2 = \frac{1(1 \cdot 2 + 1 \cdot 3) - 0 \cdot 1}{1^2 + 1^2} = -\frac{1}{2} \]
\[ y_2 = \frac{1(1 \cdot 3 - 1 \cdot 2) - 0 \cdot 1}{1^2 + 1^2} = \frac{1}{2} \]
Thus, the foot of the perpendicular from the focus to the second tangent is \( \left( -\frac{1}{2}, \frac{1}{2} \right) \).

### Step 3: Find the distance \( a \) from the focus to the vertex
The distance \( a \) can be calculated using the distance formula between the focus and the vertex (the midpoint of the two feet):
\[ \text{Midpoint} = \left( \frac{\frac{5}{2} - \frac{1}{2}}{2}, \frac{\frac{5}{2} + \frac{1}{2}}{2} \right) = \left( \frac{2}{2}, \frac{6}{4} \right) = (1, \frac{3}{2}) \]
Now, calculate the distance from the focus \( S(2, 3) \) to the vertex \( (1, \frac{3}{2}) \):
\[ a = \sqrt{(2 - 1)^2 + \left(3 - \frac{3}{2}\right)^2} = \sqrt{1 + \left(\frac{3}{2}\right)^2} = \sqrt{1 + \frac{9}{4}} = \sqrt{\frac{13}{4}} = \frac{\sqrt{13}}{2} \]

### Step 4: Calculate the length of the latus rectum
The length of the latus rectum \( L \) of a parabola is given by:
\[ L = 4a \]
Substituting the value of \( a \):
\[ L = 4 \cdot \frac{\sqrt{13}}{2} = 2\sqrt{13} \]

### Final Answer
The length of the latus rectum of the parabola is \( 2\sqrt{13} \).

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