Let z be a complex number such that z=1-t+ i √t2+t+2, where t ∈ R, then locus of z on the Argand plane is

Question

Let z be a complex number such that z=1-t+ i √t2+t+2, where t ∈ R, then locus of z on the Argand plane is (A) a parabola (B) an ellipse (C) a hyperbola (D) a pair of straight line.

Tardigrade Admin · Accepted Answer

Let z=x+i y, then x=1-t, y=√t2+t+2 ⇒ t=1-x text and y2=t2+t+2=(t+1 / 2)2+7 / 4 ⇒ y2=(x-3 / 2)2+7 / 4 This represents a hyperbola.

Q. Let $z$ be a complex number such that $z = 1 - t +$ $i t^{2} + t + 2$ , where $t \in R$ , then locus of $z$ on the Argand plane is

a parabola

an ellipse

a hyperbola

a pair of straight line.

Let $z = x + i y$ , then
$x = 1 - t, y = t^{2} + t + 2$
$\Rightarrow t = 1 - x and y^{2} = t^{2} + t + 2 = (t + 1/2)^{2} + 7/4$
$\Rightarrow y^{2} = (x - 3/2)^{2} + 7/4$
This represents a hyperbola.

Q. Let z be a complex number such that z=1−t+ it2+t+2​, where t∈R, then locus of z on the Argand plane is