Question

When \[\dfrac{{z + i}}{{z + 2\;}}\] is purely imaginary, the locus described by the point \[z\] in the Argand diagram is
A) Circle of radius \[\dfrac{{\sqrt 5 }}{2}\]
B) Circle of radius \[\dfrac{5}{4}\]
C) Straight line
D) Parabola

Answer 1

Hint: in this question we have to find locus described by the point \[z\] in the Argand diagram representing what shape. First write the given complex number as a combination of real and imaginary number. Put z in form of real and imaginary number into the equation.



Formula Used:\[z = x + iy\]
Where
z is a complex number
x represent real part of complex number
iy is a imaginary part of complex number
i is iota
Square of iota is equal to the negative of one



Complete step by step solution:Given: A complex number and it is given that complex number is purely imaginary
Now we have complex number\[\dfrac{{z + i}}{{z + 2\;}}\]
We know that
\[z = x + iy\]
Put this value in\[\dfrac{{z + i}}{{z + 2\;}}\]
\[\dfrac{{(x + iy) + i}}{{(x + iy) + 2\;}}\]
\[\dfrac{{x + iy + i}}{{x + iy + 2\;}} = \dfrac{{x + i(y + 1)}}{{(x + 2) + iy\;}}\]
\[\dfrac{{x + i(y + 1)}}{{(x + 2) + iy\;}} = \dfrac{{[x + i(y + 1)][(x + 2) - iy]}}{{[x + 2) + iy][(x + 2) - iy]}}\]
\[[\dfrac{{{x^2} + 2x + {y^2} + y}}{{{{(x + 2)}^2} + {y^2}}}] + i\dfrac{{(y + 1)(x + 2) - xy}}{{{{(x + 2)}^2} + {y^2}}}\]
It is given in the question that complex number is purely imaginary, so its real part will be zero
\[\dfrac{{{x^2} + 2x + {y^2} + y}}{{{{(x + 2)}^2} + {y^2}}} = 0\]
\[{x^2} + 2x + {y^2} + y = 0\]
This equation represents the circle. \[{{\bf{x}}^{\bf{2}}}\; + {\rm{ }}{{\bf{y}}^{\bf{2}}}\; + {\rm{ }}{\bf{2gx}}{\rm{ }} + {\rm{ }}{\bf{2fy}}{\rm{ }} + {\rm{ }}{\bf{c}}{\rm{ }} = {\rm{ }}{\bf{0}}\]
Radius of circle is given by
\[\sqrt {{g^2} + {f^2} - c} \]
Radius of circle is \[\sqrt {1 + \dfrac{1}{4} - 0} = \dfrac{{\sqrt 5 }}{2}\]
Here \[{x^2} + 2x + {y^2} + y = 0\]represent the equation of circle therefore locus of point represent circle.



Option ‘A’ is correct



Note: Complex number is a number which is a combination of real and imaginary number. So in combination number question we have to represent number as a combination of real and its imaginary part. Imaginary part is known as iota. Square of iota is equal to negative one.