Question

The two parabolas \( y^2 = 4a(x + c) \) and \( y^2 = 4bx \), \( a>b>0 \) cannot have a common normal unless:

• A. \( c>2(a + b) \)
• B. \( c>2(a - b) \)
• C. \( c<2(a - b) \)
• D. \( c<\frac{2}{a - b} \)

Hint:

To find the condition for non-existence of a common normal between two parabolas, analyze the geometry and use normal equations strategically.

Answer 1

The Correct Option is B

Step 1: Understand the condition for common normal.
If a common normal exists to two parabolas, then the normals from a point on one parabola must intersect the second parabola.
Step 2: Geometric interpretation.
The parabola \( y^2 = 4a(x + c) \) is a right-shifted version of \( y^2 = 4ax \).
To avoid a common normal, the horizontal shift \( c \) must exceed a critical value.
Step 3: Result from geometry.
This condition is \( c>2(a - b) \) to ensure no common normal exists.