Question

The equation\[\dfrac{{\left| {z - 5i} \right|}}{{\left| {z + 5i} \right|}} = 12\], where\[z = x + iy\], represents a/an [AMU\[1999\]]
A) Circle
B) Ellipse
C) Parabola
D) No real curve

Answer 1

Hint: in this question we have to find which mathematical shape the given equation of complex number represents. 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 complex number equation\[\dfrac{{\left| {z - 5i} \right|}}{{\left| {z + 5i} \right|}} = 12\]
We know that complex numbers are written as a combination of real and imaginary numbers.
\[z = x + iy\]
Where
z is a complex number
x represent real part of complex number

iy is a imaginary part of complex number
Put this value in\[\dfrac{{\left| {z - 5i} \right|}}{{\left| {z + 5i} \right|}} = 12\]
Now we get
\[\begin{array}{l}\left| {(x + iy) - 5i} \right| = 12\left| {(x + iy) + 5i} \right|\\\end{array}\]
We know that
\[\left| z \right| = \sqrt {{x^2} + {y^2}} \]
\[\left| {{x^2} + {{(y - 5)}^2}} \right| = 12\left| {{x^2} + {{(y + 5)}^2}} \right|\]
\[{x^2} + {y^2} + 25 - 10y = 12[{x^2} + {y^2} + 25 + 10y]\]
Required equation is
\[11{x^2} + 11{y^2} + 130y + 275 = 0\]
This equation represents the equation of a circle.



Option ‘A’ 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 the number as a combination of real and its imaginary part. Imaginary part is known as iota. Square of iota is equal to the negative one.