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Question: Let z = 1 – t + i\(\sqrt{t^{2} + t + 2}\), where t is a real parameter. The locus of z in the Argan...

Let z = 1 – t + it2+t+2\sqrt{t^{2} + t + 2}, where t is a real parameter. The

locus of z in the Argand plane is –

A

A straight line

B

A hyperbola

C

An ellipse

D

None of these

Answer

A hyperbola

Explanation

Solution

Sol. x + iy = 1 – t + t2+t+2\sqrt{t^{2} + t + 2}

x = 1 – t, y = t2+t+2\sqrt{t^{2} + t + 2}

t = 1 – x y2 = t2+t+2t^{2} + t + 2

y2 = (1 – x)2 + (1 – x) + 2

y2 – (x−32)2\sqrt{\left( x - \frac{3}{2} \right)^{2}}= 74\frac{7}{4}

which is a hyperbola