The locus of z such that \(\frac{|z-i|}{|z+i|}\)= 2, where z = x+iy. is
The locus of z such that \(\frac{|z-i|}{|z+i|}\)= 2, where z = x+iy. is
3x2 + 3y2 +10y + 3
3x2 - 3y2 - 10y - 3 = 0
3x2 + 3y2 + 10y + 3 = 0
x2 + y2 - 5y + 3 = 0
The Correct Option is C
Solution and Explanation
The correct option is: (A): 3x2 + 3y2 + 10y + 3 = 0.
Given: ∣z−i∣=2∣z+i∣
Hence: ∣2z−i∣∣2=∣∣1z+i∣∣2
This simplifies to: x2+(y−1)2=4(x2+(y+1)2)
Further simplifying: 3x2+4(y+1)2−(y−1)2=0
And: 3x2+3y2+8y+2y+4−1=0
Finally: 3x2 + 3y2 + 10y + 3 = 0.
Top Questions on complex numbers
- If \(r = |z|,\ θ = arg(z)\) and \( z = 2 – 2\ tan (\frac {5\pi}{8})\) then find \((r, θ)\).
- JEE Main - 2024
- Mathematics
- complex numbers
- If $z$ is a complex number such that $|z| \geq 1$, then the minimum value of \[\left| z + \frac{1}{2}(3 + 4i) \right|\]is:
- JEE Main - 2024
- Mathematics
- complex numbers
- Let $z$ be a complex number such that $|z + 2| = 1$ and $\text{Im}\left(\frac{z+1}{z+2}\right) = \frac{1}{5}$. Then the value of $|\text{Re}(z+2)|$ is:
- JEE Main - 2024
- Mathematics
- complex numbers
- The sum of all possible values of \(\theta \in [-\pi, 2\pi]\), for which \[ \frac{1 + i \cos\theta}{1 - 2i \cos\theta} \] is purely imaginary, is equal to:
- JEE Main - 2024
- Mathematics
- complex numbers
- If $z_1, z_2$ are two distinct complex numbers such that \[ \frac{|z_1 - 2z_2|}{\left| \frac{1}{2} - z_1 \overline{z_2} \right|} = 2, \] then
- JEE Main - 2024
- Mathematics
- complex numbers
Questions Asked in TS EAMCET exam
The ratio of the radii of two solid spheres of same mass in 2:3. The ratio of the moments of inertia of the spheres about their diameters is:
- TS EAMCET - 2023
- rotational motion
If (-c, c) is the set of all values of x for which the expansion is (7 - 5x)-2/3 is valid, then 5c + 7 =
- TS EAMCET - 2023
- Algebra of Complex Numbers
The general solution of the differential equation (x2 + 2)dy +2xydx = ex(x2+2)dx is
- TS EAMCET - 2023
- Differential equations
If i=√-1 then
\[Arg\left[ \frac{(1+i)^{2025}}{1+i^{2022}} \right] =\]- TS EAMCET - 2023
- Trigonometric Identities
If nCr denotes the number of combinations of n distinct things taken r at a time, then the domain of the function g (x)= (16-x)C(2x-1) is
- TS EAMCET - 2023
- permutations and combinations
Concepts Used:
Complex Number
A Complex Number is written in the form
a + ib
where,
- “a” is a real number
- “b” is an imaginary number
The Complex Number consists of a symbol “i” which satisfies the condition i^2 = −1. Complex Numbers are mentioned as the extension of one-dimensional number lines. In a complex plane, a Complex Number indicated as a + bi is usually represented in the form of the point (a, b). We have to pay attention that a Complex Number with absolutely no real part, such as – i, -5i, etc, is called purely imaginary. Also, a Complex Number with perfectly no imaginary part is known as a real number.