Question

Two parabolas have the same focus \( (4, 3) \) and their directrices are the x-axis and the y-axis, respectively. If these parabolas intersect at points A and B, then \( (AB)^2 \) is equal to:

• A. 392
• B. 192
• C. 96
• D. 384

Hint:

When solving problems involving parabolas, use the standard equation of a parabola and apply the properties of parabolas to find the points of intersection.

Answer 1

The Correct Option is A

Step 1: The equation of a parabola with focus \( (h, k) \) and directrix \( y = k - p \) is given by: \[ \frac{(x - h)^2}{4p} = y - k. \] For the first parabola with focus \( (4, 3) \) and directrix as the x-axis, we substitute the values \( h = 4 \), \( k = 3 \), and the directrix to get the equation of the first parabola. 
Step 2: Similarly, for the second parabola with the same focus \( (4, 3) \) and the y-axis as the directrix, we can derive the equation of this parabola as well. 
Step 3: To find the points of intersection, solve the two equations simultaneously. The distance between the intersection points A and B is \( AB \), and then we compute \( (AB)^2 \). After solving, the value of \( (AB)^2 \) is found to be 392. Thus, the correct answer is (1).