Advertisements
Advertisements
Question
The equation |z + 1 – i| = |z – 1 + i| represents a ______.
Options
Straight line
Circle
Parabola
Hyperbola
Solution
The equation |z + 1 – i| = |z – 1 + i| represents a straight line.
Explanation:
|z + 1 – i| = |z – 1 + i|
⇒ |z – (–1 + i)| = |z – (1 – i)|
⇒ PA = PB Where A denotes the point (–1, 1), B denotes the point (1, –1) and P denotes the point (x, y).
⇒ z lies on the perpendicular bisector of the line joining A and B and perpendicular bisector is a straight line.
APPEARS IN
RELATED QUESTIONS
If `x – iy = sqrt((a-ib)/(c - id))` prove that `(x^2 + y^2) = (a^2 + b^2)/(c^2 + d^2)`
Find the value of i49 + i68 + i89 + i110
Simplify the following and express in the form a + ib:
`5/2"i"(- 4 - 3 "i")`
Simplify the following and express in the form a + ib:
(1 + 3i)2 (3 + i)
Write the conjugates of the following complex number:
`-sqrt(5) - sqrt(7)"i"`
Find the value of 1 + i2 + i4 + i6 + i8 + ... + i20
Evaluate : `("i"^37 + 1/"i"^67)`
Prove that `(1 + "i")^4 xx (1 + 1/"i")^4` = 16
If (x + iy)3 = y + vi then show that `(y/x + "v"/y)` = 4(x2 – y2)
Find the value of x and y which satisfy the following equation (x, y∈R).
(x + 2y) + (2x − 3y)i + 4i = 5
Answer the following:
Simplify the following and express in the form a + ib:
(2i3)2
Answer the following:
Solve the following equation for x, y ∈ R:
(4 − 5i)x + (2 + 3i)y = 10 − 7i
Answer the following:
Find the value of x4 + 9x3 + 35x2 − x + 164, if x = −5 + 4i
Answer the following:
Show that `(1/sqrt(2) + "i"/sqrt(2))^10 + (1/sqrt(2) - "i"/sqrt(2))^10` = 0
If z ≠ 1 and `"z"^2/("z - 1")` is real, then the point represented by the complex number z lies ______.
If |z1| = 1, |z2| = 2, |z3| = 3 and |9z1z2 + 4z1z3 + z2z3| = 12, then the value of |z1 + z2 + z3| is
If z1 = 5 + 3i and z2 = 2 - 4i, then z1 + z2 = ______.
Find the value of 2x4 + 5x3 + 7x2 – x + 41, when x = `-2 - sqrt(3)"i"`.
State true or false for the following:
The argument of the complex number z = `(1 + i sqrt(3))(1 + i)(cos theta + i sin theta)` is `(7pi)/12 + theta`.
What is the locus of z, if amplitude of z – 2 – 3i is `pi/4`?
If |z1| = 1(z1 ≠ –1) and z2 = `(z_1 - 1)/(z_1 + 1)`, then show that the real part of z2 is zero.
If |z1| = |z2| = ... = |zn| = 1, then show that |z1 + z2 + z3 + ... + zn| = `|1/z_1 + 1/z_2 + 1/z_3 + ... + 1/z_n|`.
Multiplicative inverse of 1 + i is ______.
Where does z lie, if `|(z - 5i)/(z + 5i)|` = 1.
Let x, y ∈ R, then x + iy is a non-real complex number if ______.
If z1, z2, z3 are complex numbers such that |z1| = |z2| = |z3| = `|1/z_1 + 1/z_2 + 1/z_3|` = 1, then |z1 + z2 + z3| is ______.
If α and β are the roots of the equation x2 + 2x + 4 = 0, then `1/α^3 + 1/β^3` is equal to ______.
Find the value of `(i^592+i^590+i^588+i^586+i^584)/(i^582+i^580+i^578+i^576+i^574)`
Simplify the following and express in the form a+ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18`
