Question

The length of the latus rectum of the parabola whose focus is at \( (1,-2) \) and directrix is the line \( x + y + 3 = 0 \) is

• A. \( 8\sqrt{2} \) units
• B. \( 2\sqrt{2} \) units
• C. \( \sqrt{2} \) units
• D. \( 4\sqrt{2} \) units

Hint:

For any parabola, the length of the latus rectum is always equal to \(4a\).

Answer 1

The Correct Option is B

Step 1: Find the distance between focus and directrix.
Distance of focus \( (1,-2) \) from the directrix \( x+y+3=0 \) is \[ d = \frac{|1-2+3|}{\sqrt{1^2+1^2}} = \frac{2}{\sqrt{2}} = \sqrt{2} \]

Step 2: Determine the focal length \( a \).
For a parabola, the distance between focus and directrix equals \( 2a \). \[ 2a = \sqrt{2} \Rightarrow a = \frac{\sqrt{2}}{2} \]

Step 3: Use the formula for length of latus rectum.
\[ \text{Length of latus rectum} = 4a = 4 \times \frac{\sqrt{2}}{2} = 2\sqrt{2} \]

Step 4: Conclusion.
\[ \boxed{2\sqrt{2} \text{ units}} \]