JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations Question Bank Mock Test - Complex Numbers and Quadratic Equations

Let \[z=1-t+i \sqrt{{{t}^{2}}+t+2}\], where t is a real parameter. The locus of z in the argand plane is

A) a hyperbola

B) an ellipse

C) a straight line

D) none of these

Correct Answer: A

Solution :

[a] \[x+iy=1-t+i\sqrt{{{t}^{2}}+t+2}\]

\[\Rightarrow x=1-t,y=\sqrt{{{t}^{2}}+t+2}\]

Eliminating t, \[{{y}^{2}}={{t}^{2}}+t+2={{(1-x)}^{2}}+1-x+2={{\left( x-\frac{3}{2} \right)}^{2}}+\frac{7}{4}\]

\[\Rightarrow {{y}^{2}}-{{\left( x-\frac{3}{2} \right)}^{2}}=\frac{7}{4}\], which is a hyperbola