Question

The maximum number of common normals of \( y^2 = 4ax \) and \( x^2 = 4by \) is:

• A. \( 3 \)
• B. \( 4 \)
• C. \( 5 \)
• D. \( 6 \)

Hint:

Common normals of conics: \begin{itemize} \item Parameterize normals. \item Solve resulting algebraic system. \item Degree product gives max count. \end{itemize}

Answer 1

The Correct Option is D

Concept: Normal to parabola \( y^2 = 4ax \) at parameter \( t \): \[ y = -tx + 2at + at^3 \] Normal to \( x^2 = 4by \): \[ x = -sy + 2bs + bs^3 \] Step 1: {\color{red}Equate slopes.} For common normal: \[ -t = \frac{-1}{s} \Rightarrow ts = 1 \] Step 2: {\color{red}Substitute relation.} Remaining equations lead to cubic in parameter. Two cubics intersect ⇒ up to 6 solutions. Step 3: {\color{red}Maximum count.} Maximum common normals = 6.