Q96 / 155 · WBJEE 2025 · Mathematics

The maximum number of common normals of $y^2=4 a x$ and $x^2=4 b y$ is equal to

  • A
    $3$
  • B
    $4$
  • Correct answer
    C
    $5$
  • D
    $6$

Solution

$\mathrm{y}^2=4 \mathrm{ax} \text { and } \mathrm{x}^2=4 \mathrm{by}$ Eq. of normal to $y^2=4 a x$ and $x^2=4 b y$ are $y=m x-2 a m-a m^3$ and $y=m x+2 b+\frac{b}{m^2}$ For common normal; $2 \mathrm{~b}+\frac{\mathrm{b}}{\mathrm{m}^2}+2 \mathrm{am}+\mathrm{am}^3=0$ $\Rightarrow a m^5+2 a m^3+2 b m^2+b=0$ Max. 5 common normal

Hence, the correct answer is option: 'C'.

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