Question

If the vertex of a parabola is \( (2, -1) \) and the equation of its directrix is \[ 4x - 3y = 21, \] then the length of its latus rectum is:

• A. 2
• B. 8
• C. 12
• D. 16

Hint:

The latus rectum of a parabola is given by \( 4a \), where \( a \) is the focal distance from the vertex.

Answer 1

The Correct Option is B

Step 1: Finding the focus. 
The focus lies on the perpendicular bisector of the vertex and the directrix. Using the perpendicular distance formula, \[ \frac{|4(2) - 3(-1) - 21|}{\sqrt{4^2 + (-3)^2}} = \frac{|8 + 3 - 21|}{5} = \frac{10}{5} = 2 \] So, the focal distance is 2. 
Step 2: Finding the latus rectum. 
The length of the latus rectum is given by: \[ \frac{4a}{|m|} \] where \( a = 2 \), giving \[ \frac{4(2)}{1} = 8 \] Thus, the correct answer is (B).