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.
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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.
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