Advertisements
Advertisements
प्रश्न
If |z1| = 1(z1 ≠ –1) and z2 = `(z_1 - 1)/(z_1 + 1)`, then show that the real part of z2 is zero.
Advertisements
उत्तर
Let z1 = x + yi
|z1| = `sqrt(x^2 + y^2)` = 1 ......[Given that |z1| = 1]
⇒ x2 + y2 = 1 ......(i)
Now z2 = `(z_1 - 1)/(z_1 + 1)`
= `(x + yi - 1)/(x + yi + 1)`
= `((x + 1) + y"i")/((x + 1) + y"i")`
= `((x - 1) + yi)/((x + 1) + yi) xx (x + 1 - yi)/(x + 1 - yi)`
= `((x - 1)(x + 1) - y(x - 1)i + y(x + 1)i - y^2i^2)/((x + 1)^2 - y^2i^2)`
= `(x^2 - 1 + yi(x + 1 - x + 1) + y^2)/(x^2 + 1 + 2x + y^2)`
= `((x^2 + y^2 - 1) + 2yi)/(x^2 + y^2 + 2x + 1)`
= `((1 - 1))/(x^2 + y^2 + 2x + 1) + (2y)/(x^2 + y^2 + 2x + 1) "i"`
= `0 + (2y)/(x^2 + y^2 + 2x + 1) "i"`
Hence, the real part of z2 is 0.
APPEARS IN
संबंधित प्रश्न
Find the multiplicative inverse of the complex number.
–i
If (a + ib) (c + id) (e + if) (g + ih) = A + iB, then show that (a2 + b2) (c2 + d2) (e2 + f2) (g2 + h2) = A2 + B2.
Find the value of: x3 – x2 + x + 46, if x = 2 + 3i
Simplify the following and express in the form a + ib:
`(sqrt(5) + sqrt(3)"i")/(sqrt(5) - sqrt(3)"i")`
Write the conjugates of the following complex number:
`sqrt(5) - "i"`
Write the conjugates of the following complex number:
`sqrt(2) + sqrt(3)"i"`
Find the value of i + i2 + i3 + i4
Simplify:
`(i^592 + i^590 + i^588 + i^586 + i^584)/(i^582 + i^580 + i^578 + i^576 + i^574)`
Find the value of x and y which satisfy the following equation (x, y∈R).
If x + 2i + 15i6y = 7x + i3 (y + 4), find x + y
Select the correct answer from the given alternatives:
The value of is `("i"^592 + "i"^590 + "i"^588 + "i"^586 + "i"^584)/("i"^582 + "i"^580 + "i"^578 + "i"^576 + "i"^574)` is equal to:
Answer the following:
Simplify the following and express in the form a + ib:
(1 + 3i)2(3 + i)
Answer the following:
Simplify the following and express in the form a + ib:
`(1 + 2/"i")(3 + 4/"i")(5 + "i")^-1`
Answer the following:
Solve the following equations for x, y ∈ R:
(x + iy) (5 + 6i) = 2 + 3i
Answer the following:
show that `((1 + "i")/sqrt(2))^8 + ((1 - "i")/sqrt(2))^8` = 2
Answer the following:
Show that z = `((-1 + sqrt(-3))/2)^3` is a rational number
If |z1| = 1, |z2| = 2, |z3| = 3 and |9z1z2 + 4z1z3 + z2z3| = 12, then the value of |z1 + z2 + z3| is
The argument of the complex number `(4 + 9i)/(13 + 5i)` is ______
The value of (2 + i)3 × (2 – i)3 is ______.
Evaluate: (1 + i)6 + (1 – i)3
Locate the points for which 3 < |z| < 4.
Find the value of 2x4 + 5x3 + 7x2 – x + 41, when x = `-2 - sqrt(3)"i"`.
Find the value of k if for the complex numbers z1 and z2, `|1 - barz_1z_2|^2 - |z_1 - z_2|^2 = k(1 - |z_1|^2)(1 - |"z"_2|^2)`
What is the locus of z, if amplitude of z – 2 – 3i is `pi/4`?
The equation |z + 1 – i| = |z – 1 + i| represents a ______.
Number of solutions of the equation z2 + |z|2 = 0 is ______.
If `(1 + i)^2/(2 - i)` = x + iy, then find the value of x + y.
If (1 + i)z = `(1 - i)barz`, then show that z = `-ibarz`.
Solve the equation |z| = z + 1 + 2i.
The value of `sqrt(-25) xx sqrt(-9)` is ______.
The number `(1 - i)^3/(1 - i^2)` is equal to ______.
Find `|(1 + i) ((2 + i))/((3 + i))|`.
If `((1 + i)/(1 - i))^x` = 1, then ______.
The complex number z which satisfies the condition `|(i + z)/(i - z)|` = 1 lies on ______.
`((1 + cosθ + isinθ)/(1 + cosθ - isinθ))^n` = ______.
If a complex number z satisfies the equation `z + sqrt(2)|z + 1| + i` = 0, then |z| is equal to ______.
Simplify the following and express in the form a+ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
Show that `(-1 + sqrt3i)^3` is a real number.
