Latus rectum of the parabola whose focus is (3, 4) and whose tangent at vertex has the equation `x + y = 7 + 5sqrt2` is

Options

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

Answer

To find the latus rectum of the parabola with a given focus and tangent at the vertex, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Information:**
- Focus of the parabola: \( (3, 4) \)
- Equation of the tangent at the vertex: \( x + y = 7 + 5\sqrt{2} \)

2. **Convert the Tangent Equation:**
- Rewrite the tangent equation in the standard form:
\[ x + y - (7 + 5\sqrt{2}) = 0 \]
- Here, \( a = 1 \), \( b = 1 \), and \( c = -(7 + 5\sqrt{2}) \).

3. **Use the Distance Formula:**
- The distance \( d \) from a point \( (x_1, y_1) \) to a line \( ax + by + c = 0 \) is given by:
\[ d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} \]
- Substitute \( (x_1, y_1) = (3, 4) \) into the formula:
\[ d = \frac{|1 \cdot 3 + 1 \cdot 4 - (7 + 5\sqrt{2})|}{\sqrt{1^2 + 1^2}} \]

4. **Calculate the Numerator:**
- Calculate \( 3 + 4 - (7 + 5\sqrt{2}) \):
\[ 3 + 4 = 7 \quad \text{so} \quad 7 - (7 + 5\sqrt{2}) = -5\sqrt{2} \]
- Therefore, the absolute value is:
\[ | -5\sqrt{2} | = 5\sqrt{2} \]

5. **Calculate the Denominator:**
- The denominator is:
\[ \sqrt{1^2 + 1^2} = \sqrt{2} \]

6. **Combine to Find the Distance:**
- Now substitute back into the distance formula:
\[ d = \frac{5\sqrt{2}}{\sqrt{2}} = 5 \]
- This distance \( d \) represents \( a \) (the distance from the vertex to the focus).

7. **Calculate the Length of the Latus Rectum:**
- The length of the latus rectum \( L \) is given by:
\[ L = 4a = 4 \times 5 = 20 \]

8. **Final Answer:**
- The length of the latus rectum of the parabola is \( 20 \).