Q262 / 270 · NDA 2015 (Phase 1) · Mathematics

Consider the parametric equation $x=\frac{a\left(1-t^2\right)}{1+t^2}, y=\frac{2 a t}{1+t^2}$ What does the equation represent?

  • A
    It represents a circle of diameter a
  • Correct answer
    B
    It represents a circle of radius a
  • C
    It represents a parabola
  • D
    None of the above

Solution

$\begin{aligned} & x^2+y^2=\left[\frac{a\left(1-t^2\right)}{1+t^2}\right]^2+\left[\frac{2 a t}{1+t^2}\right]^2 \\ & =\frac{a^2\left(1-t^2\right)^2}{\left(1+t^2\right)^2}+\frac{4 a^2 t^2}{\left(1+t^2\right)^2} \\ & =\frac{a^2\left(1+t^4-2 t^2\right)+4 a^2 t^2}{\left(1+t^2\right)^2} \\ & =\frac{a^2\left(1+t^4+2 t^2\right)}{\left(1+t^2\right)^2}=\frac{a^2\left(1+t^2\right)^2}{\left(1+t^2\right)^2}=a^2\end{aligned}$ Hence, the given equation represent a circle of radius (a) $\therefore \quad$ Option (B) is correct.

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

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