Question

Let $Z \in C$ , then what does the equation $2\left| {z + 3i} \right| - \left| {z - i} \right| = 0$ represent?
Let $Z \in C$ , the set of complex numbers. Then the equation $2\left| {z + 3i} \right| - \left| {z - i} \right| = 0$ represents
A) A circle with radius $\dfrac{8}{3}$ .
B) A circle with diameter $\dfrac{{10}}{3}$ .
C) An ellipse with length of minor axis $\dfrac{{16}}{9}$
D) An ellipse with length of major axis $\dfrac{{16}}{3}$

Answer 1

Hint : As we know that complex numbers are those numbers that consist of two parts i.e. a real number and an imaginary number. The standard form of complex number is $a + ib$ where $a$ is the real number and the second part i.e. $ib$ is the imaginary number.

Complete step by step solution:
As per the question we have $2\left| {z + 3i} \right| - \left| {z - i} \right| = 0$ . We will substitute the value of $z = x + iy$ and the given equation of circle can be written as
$2\left| {x + iy + 3i} \right| - \left| {x + iy - i} \right| = 0$ .
It can be written as
$2\left| {x + (y + 3)i} \right| - \left| {x + (y - 1)i} \right| = 0 \\\ \Rightarrow 2\left| {x + (y + 3)i} \right| = \left| {x + (y - 1)i} \right|\;$ .
Now by squaring we have: $\left| {x + iy} \right| = \sqrt {{x^2} + {y^2}}$ .
By substituting the values: $2\sqrt {{x^2} + {{(y + 3)}^2}} = \sqrt {{x^2} + {{(y - 1)}^2}}$ .
We will solve it now
$4{x^2} + 4(y + 3) = {x^2} + {(y - 1)^2}\\\ \Rightarrow 4{x^2} + 4{y^2} + 36 + 24y = {x^2} + {y^2} + 1 - 2y\;$ .
By transferring all the values in the left hand side of the equation:
$4{x^2} + 4{y^2} + 36 + 24y - {x^2} - {y^2} - 1 + 2y = 0\\\ \Rightarrow 3{x^2} + 3{y^2} + 26y + 35 = 0\;$ .
Dividing the equation by $3$ we get:
${x^2} + {y^2} + \dfrac{{26y}}{{3}} + \dfrac{{35}}{3} = 0$ .
We know that the general form of the equation is
${x^2} + {y^2} + 2gx + 2fy + c = 0$ .
We know that the centre and radius of circle
${x^2} + {y^2} + 2gx + 2fy + c = 0$ is defined as $( - g, - f)$ and so the formula of radius is $\sqrt {{g^2} + {f^2} - c}$ .
Therefore the radius of the circle is
$r = \sqrt {{{\left( {\dfrac{{13}}{3}} \right)}^2} - \dfrac{{35}}{3} + 0}$ . It gives us $\sqrt {\dfrac{{169 - 105}}{9}} = \sqrt {\dfrac{{64}}{9}}$ . So the required value is $\dfrac{8}{3}$ .
Hence the correct answer is (a) A circle with radius $\dfrac{8}{3}$ .
So, the correct answer is “Option A”.

Note : We should note that in the above question sum of squares formula is used i.e. ${(a + b)^2} = {a^2} + {b^2} + 2ab$ and another one is difference of square formula i.e. ${(a - b)^2} = {a^2} + {b^2} - 2ab$ . Before solving this kind of question we should know the equation of circle and the radius of the circle. The formula of radius is $r = \sqrt {{{(x - h)}^2} + {{(y - k)}^2}}$ .