JEE Advanced 2016 · Paper 2, Mathematics question 47

Real and imaginary parts.
$$z=\frac{a-ibt}{a^2+b^2t^2},\qquad x=\frac a{a^2+b^2t^2},\quad y=\frac{-bt}{a^2+b^2t^2}.$$

Locus. $x^2+y^2=\dfrac1{a^2+b^2t^2}=\dfrac xa$.

• (A) For $a>0$, $b\ne0$: the circle with centre $\left(\frac1{2a},0\right)$ and radius $\frac1{2a}$. True.
• (B) For $a<0$, the centre is $\left(\frac1{2a},0\right)$, not $\left(-\frac1{2a},0\right)$. False.
• (C) $b=0$: $z=1/a$ is real, on the $x$-axis. True.
• (D) $a=0$: $z=-i/(bt)$ is purely imaginary, on the $y$-axis. True.

(A), (C), (D)