Advertisements
Advertisements
Question
If `x – iy = sqrt((a-ib)/(c - id))` prove that `(x^2 + y^2) = (a^2 + b^2)/(c^2 + d^2)`
Advertisements
Solution
`x – iy = sqrt((a-ib)/(c - id))` ...... (1)
In place of i - on writing, i
`x – iy = sqrt((a-ib)/(c - id))`
On multiplying equations (1) and (2), we get
`(x - iy)(x + iy) = sqrt((a - ib)/(c - id)) xx sqrt((a + ib)/(c + id))`
= or `x^2 - i^2y^2 = sqrt((a^2 - i^2 b^2)/(c^2 - i^2 d^2))`
∴ `x^2 + y^2 = sqrt((a^2 + b^2)/(c^2 + d^2)`
On squaring both sides,
`(x^2 + y^2)^2 = (a^2 + b^2)/(c^2 + d^2)`
APPEARS IN
RELATED QUESTIONS
Find the multiplicative inverse of the complex number.
–i
Express the following expression in the form of a + ib.
`((3 + sqrt5)(3 - isqrt5))/((sqrt3 + sqrt2i)-(sqrt3 - isqrt2))`
Find the value of: 2x3 – 11x2 + 44x + 27, if x = `25/(3 - 4"i")`
Find the value of: x3 – 3x2 + 19x – 20, if x = 1 – 4i
Find the value of i49 + i68 + i89 + i110
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)`
If x + iy = (a + ib)3, show that `x/"a" + y/"b"` = 4(a2 − b2)
Show that `((sqrt(7) + "i"sqrt(3))/(sqrt(7) - "i"sqrt(3)) + (sqrt(7) - "i"sqrt(3))/(sqrt(7) + "i"sqrt(3)))` is real
Find the value of x and y which satisfy the following equation (x, y ∈ R).
`((x + "i"y))/(2 + 3"i") + (2 + "i")/(2 - 3"i") = 9/13(1 + "i")`
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:
(2i3)2
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:
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
Answer the following:
Evaluate: i131 + i49
Answer the following:
Find the value of x4 + 9x3 + 35x2 − x + 164, if x = −5 + 4i
Answer the following:
If x + iy = `("a" + "ib")/("a" - "ib")`, prove that x2 + y2 = 1
If z1 = 2 – 4i and z2 = 1 + 2i, then `bar"z"_1 + bar"z"_2` = ______.
The value of (2 + i)3 × (2 – i)3 is ______.
Find the value of 2x4 + 5x3 + 7x2 – x + 41, when x = `-2 - sqrt(3)"i"`.
State true or false for the following:
The complex number cosθ + isinθ can be zero for some θ.
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`.
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.
What is the smallest positive integer n, for which (1 + i)2n = (1 – i)2n?
1 + i2 + i4 + i6 + ... + i2n is ______.
If (1 + i)z = `(1 - i)barz`, then show that z = `-ibarz`.
If |z1| = |z2| = ... = |zn| = 1, then show that |z1 + z2 + z3 + ... + zn| = `|1/z_1 + 1/z_2 + 1/z_3 + ... + 1/z_n|`.
The value of `sqrt(-25) xx sqrt(-9)` is ______.
If |z + 4| ≤ 3, then the greatest and least values of |z + 1| are ______ and ______.
State True or False for the following:
The inequality |z – 4| < |z – 2| represents the region given by x > 3.
Find `|(1 + i) ((2 + i))/((3 + i))|`.
Where does z lie, if `|(z - 5i)/(z + 5i)|` = 1.
Let x, y ∈ R, then x + iy is a non-real complex number if ______.
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)`
Find the value of `sqrt(-3) xx sqrt(-6)`
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.
