Question

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

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

Hint:

When finding common normals to two conic sections, solving for the slopes of the normals from both curves typically gives the number of common normals. The number of solutions depends on the geometry and relative positioning of the curves.

Answer 1

The Correct Option is C

We analyze two parabolas:
\( y^2 = 4ax \), opening rightward.
\( x^2 = 4by \), opening upward.
The goal is to find the maximum number of shared normals.
Step 1: Normal to \( y^2 = 4ax \).
The normal to \( y^2 = 4ax \) at \( (x_1, y_1) \) is: \[\ny - y_1 = -\frac{x_1}{y_1} (x - x_1).\n\] The slope is \( -\frac{x_1}{y_1} \).
Step 2: Normal to \( x^2 = 4by \).
The normal to \( x^2 = 4by \) at \( (x_2, y_2) \) is: \[\ny - y_2 = \frac{y_2}{x_2} (x - x_2).\n\] The slope is \( \frac{y_2}{x_2} \).
Step 3: Condition for shared normals.
Shared normals have equal slopes: \[\n-\frac{x_1}{y_1} = \frac{y_2}{x_2}.\n\] This leads to equations with potentially multiple solutions. Considering all cases, the maximum number of solutions is 5.
Step 4: Conclusion.
The maximum number of common normals is 5.