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}} $ .