Q75 / 180 · COMEDK 2013 · Mathematics

If tangent to the curve $x=a t^{2}, y=2 a t$ is perpendicular to $X$-axis, then its point of contact is

  • A
    $(a, a)$
  • B
    $(0, a)$
  • Correct answer
    C
    $(0,0)$
  • D
    $(a, 0)$

Solution

We have, $$ x=a t^{2}, y=2 a t $$ Now, $\quad \frac{d x}{d t}=2 a t, \frac{d y}{d t}=2 a$ $$ \therefore \quad \frac{d y}{d x}=\frac{d y / d t}{d x / d t}=\frac{2 a}{2 a t}=\frac{1}{t} $$ Since, tangent is perpendicular to $X$-axis, then $\frac{d y}{d x}=\frac{1}{0}$ $\Rightarrow \quad \frac{1}{t}=\frac{1}{0} \Rightarrow t=0$ $\therefore \quad x=a t^{2}=0$ and $y=2 a t=0$ So, required point is $(0,0)$.

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

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