The equation of the latus rectum of a parabola is `x+y=8` and the equation of the tangent at the vertex is `x+y=12.` Then find the length of the latus rectum.

Explanation

To solve the problem, we need to find the length of the latus rectum of the parabola given the equations of the latus rectum and the tangent at the vertex.

### Step 1: Identify the equations
The equations given are:
1. Equation of the latus rectum: \( x + y = 8 \)
2. Equation of the tangent at the vertex: \( x + y = 12 \)

### Step 2: Rewrite the equations in standard form
We can rewrite these equations in the form \( Ax + By + C = 0 \):
1. For the latus rectum: \( x + y - 8 = 0 \)
2. For the tangent: \( x + y - 12 = 0 \)

### Step 3: Determine the distance between the two parallel lines
The distance \( d \) between two parallel lines of the form \( Ax + By + C_1 = 0 \) and \( Ax + By + C_2 = 0 \) is given by the formula:
\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]
Here, \( A = 1 \), \( B = 1 \), \( C_1 = -8 \), and \( C_2 = -12 \).

### Step 4: Substitute the values into the formula
Substituting the values into the distance formula:
\[ d = \frac{|-12 - (-8)|}{\sqrt{1^2 + 1^2}} = \frac{|-12 + 8|}{\sqrt{2}} = \frac{|-4|}{\sqrt{2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2} \]

### Step 5: Relate the distance to the length of the latus rectum
The length of the latus rectum \( L \) of a parabola is given by the formula:
\[ L = 4p \]
where \( p \) is the distance from the vertex to the focus. The distance we calculated \( d \) represents \( 2p \) (the distance between the latus rectum and the vertex). Therefore:
\[ 2p = 2\sqrt{2} \implies p = \sqrt{2} \]
Thus, the length of the latus rectum is:
\[ L = 4p = 4(\sqrt{2}) = 4\sqrt{2} \]

### Final Answer
The length of the latus rectum is \( 4\sqrt{2} \) units.

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