Question

If \[\left| {z - 2 - 3i} \right| + \left| {z + 2 - 6i} \right| = 4\], where \[i = \sqrt { - 1} \], then locus of P(z) is [DCE \[2005\]]
A) An ellipse
B) \[\phi \]
C) Line segment joining point \[2 + 3i\] and \[ - 2 + 6i\]
D) None of these

Answer 1

Hint: in this question we have to find the locus of points that satisfy the given equation. First, write the given complex number as a combination of real and imaginary numbers. Put z in the form of real and imaginary numbers into the equation.

Formula used: Equation of complex number is given by
\[z = x + iy\]
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: Equation in the form of complex number
Now we have equation in form of complex number\[\left| {z - 2 - 3i} \right| + \left| {z + 2 - 6i} \right| = 4\]
We know that complex numbers are written as a combination of real and imaginary numbers.
\[z = x + iy\]
Put this value in\[\left| {z - 2 - 3i} \right| + \left| {z + 2 - 6i} \right| = 4\]
\[\left| {(x + iy) - 2 - 3i} \right| + \left| {(x + iy) + 2 - 6i} \right| = 4\]
\[\left| {(x - 2) + i(y - 3)} \right| + \left| {(x + 2) + i(y - 6)} \right| = 4\]
We know that
\[\left| z \right| = \sqrt {{x^2} + {y^2}} \]
\[\sqrt {{{(x - 2)}^2} + {{(y - 3)}^2}} + \sqrt {{{(x + 2)}^2} + {{(y - 6)}^2}} = 4\]
\[\sqrt {{{(x - 2)}^2} + {{(y - 3)}^2}} = 4 - \sqrt {{{(x + 2)}^2} + {{(y - 6)}^2}} \]
Squaring both sides
\[{(x - 2)^2} + {(y - 3)^2} = {(4 - \sqrt {{{(x + 2)}^2} + {{(y - 6)}^2}} )^2}\]
\[{x^2} + 4 - 4x + {y^2} + 9 - 6y = 16 + {(x + 2)^2} + {(y - 6)^2} - 8\sqrt {{{(x + 2)}^2} + {{(y - 6)}^2}} \]
\[{x^2} + 4 - 4x + {y^2} + 9 - 6y = 16 + {x^2} + 4 + 4x + {y^2} + 36 - 12y - 8\sqrt {{{(x + 2)}^2} + {{(y - 6)}^2}} \]
\[ - 4x + 9 - 6y = 16 + 4x + 36 - 12y - 8\sqrt {{{(x + 2)}^2} + {{(y - 6)}^2}} \]
\[ - 8x - 43 + 6y = - 8\sqrt {{{(x + 2)}^2} + {{(y - 6)}^2}} \]
Squaring both sides we get
\[28{y^2} + 96xy - 432x - 252y + 711 = 0\]
This equation represents a parabola.

Thus, Option (D) is correct.

Note: Complex number is a number which is a combination of real and imaginary numbers. So in complex number questions, we have to represent numbers as a union of real and imaginary parts. Imaginary part is known as iota. Square of iota is equal to the negative one.