Advertisements
Advertisements
प्रश्न
Find the complex number satisfying the equation `z + sqrt(2) |(z + 1)| + i` = 0.
उत्तर
Given that: z + `sqrt(2) |(z + 1)| + i` = 0
Let z = x + yi
∴ `(x + yi) + sqrt(2)|(x + yi + 1)| + i` = 0
⇒ `x + (y + 1)i + sqrt(2)|(x + 1) + yi|` = 0
⇒ `x + (y + 1)i + sqrt(2) sqrt((x + 1)^2 + y^2)` = 0
⇒ `x + (y + 1)i + sqrt(2) sqrt(x^2 + 2x + 1 + y^2)` = 0 + 0i
⇒ `x + sqrt(2) sqrt(x^2 + 2x + 1 + y^2)` = 0, y + 1 = 0
⇒ x = `- sqrt(2) sqrt(x^2 + 2x + 1 + y^2)` and y = –1
⇒ x2 = 2(x2 + 2x + 1 + y2)
⇒ x2 = 2x2 + 4x + 2 + 2y2
⇒ x2 + 4x + 2 + 2y2 = 0
⇒ x2 + 4x + 2 + 2(–1)2 = 0 .....[∵y = –1]
⇒ x2 + 4x + 4 = 0
⇒ (x + 2)2 = 0
⇒ x + 2 = 0
⇒ x = –2
Hence, z = x + yi = –2 – i.
APPEARS IN
संबंधित प्रश्न
Find the multiplicative inverse of the complex number.
`sqrt5 + 3i`
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of: x3 – x2 + x + 46, if x = 2 + 3i
Simplify the following and express in the form a + ib:
(2i3)2
Write the conjugates of the following complex number:
`-sqrt(5) - sqrt(7)"i"`
Write the conjugates of the following complex number:
`sqrt(5) - "i"`
Prove that `(1 + "i")^4 xx (1 + 1/"i")^4` = 16
Find the value of `("i"^6 + "i"^7 + "i"^8 + "i"^9)/("i"^2 + "i"^3)`
Select the correct answer from the given alternatives:
`sqrt(-3) sqrt(-6)` is equal to
Answer the following:
Simplify the following and express in the form a + ib:
`(3"i"^5 + 2"i"^7 + "i"^9)/("i"^6 + 2"i"^8 + 3"i"^18)`
Answer the following:
Simplify the following and express in the form a + ib:
`(5 + 7"i")/(4 + 3"i") + (5 + 7"i")/(4 - 3"i")`
Answer the following:
Solve the following equations for x, y ∈ R:
(x + iy) (5 + 6i) = 2 + 3i
Solve the following equation for x, y ∈ R:
2x + i9y (2 + i) = xi7 + 10i16
Answer the following:
Find the value of x4 + 9x3 + 35x2 − x + 164, if x = −5 + 4i
Answer the following:
Find the real numbers x and y such that `x/(1 + 2"i") + y/(3 + 2"i") = (5 + 6"i")/(-1 + 8"i")`
Answer the following:
Simplify `[1/(1 - 2"i") + 3/(1 + "i")] [(3 + 4"i")/(2 - 4"i")]`
The argument of the complex number `(4 + 9i)/(13 + 5i)` is ______
If z1 = 2 – 4i and z2 = 1 + 2i, then `bar"z"_1 + bar"z"_2` = ______.
State true or false for the following:
If three complex numbers z1, z2 and z3 are in A.P., then they lie on a circle in the complex plane.
The area of the triangle on the complex plane formed by the complex numbers z, –iz and z + iz is ______.
State True or False for the following:
Multiplication of a non-zero complex number by –i rotates the point about origin through a right angle in the anti-clockwise direction.
The point represented by the complex number 2 – i is rotated about origin through an angle `pi/2` in the clockwise direction, the new position of point is ______.
Let |z| = |z – 3| = |z – 4i|, then the value |2z| is ______.
If `(3 + i)(z + barz) - (2 + i)(z - barz) + 14i` = 0, then `barzz` is equal to ______.
The complex number z = x + iy which satisfy the equation `|(z - 5i)/(z + 5i)|` = 1, lie on ______.
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)`
Simplify the following and express in the form a + ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
Find the value of `sqrt(-3) xx sqrt(-6)`
