Advertisements
Advertisements
प्रश्न
If z = x + iy and arg `((z - "i")/(z + 2)) = pi/4`, show that x2 + y3 + 3x – 3y + 2 = 0
उत्तर
Given z = x + iy and arg `((z - "i")/(z + 2)) = pi/4`
Now simplifying `(z - "i")/(z + 2) = (x + iy - "i")/(x + "i"y + 2)`
= `(x + "i"(y - 1))/((x + 2) + "i"y) xx ((x + 2) - "i"y)/((x + 2) - "i"y)`
= `(x(x + 2) + y(y 1))/((x + 2)^2 + y^2) + ("i"[(x + 2)(y - 1) - xy])/((x + 2)^2 + y^2)`
= `(x^2 + y^2 + 2x - y)/((x + 2)^2 + y^2) + "i" ((2y - x - 2))/((x + 2)^2 + y^2)`
Given arg `((z - "i")/(z + 2)) = pi/4`
i.e., `tan^-1 ((2y - x - 2)/(x^2 + y^2 + 2x - y)) = pi/4`
`(2y - x - 2)/(x^2 + y^2 + 2x - y) = tan pi/4` = 1
2y – x – 2 = x2 + 2x + y2 – y
x2 + y2 + 2x + x – y – 2y + 2 = 0
⇒ x² + y² + 3x – 3y + 2 = 0
Hence proved
APPEARS IN
संबंधित प्रश्न
Write in polar form of the following complex numbers
`2 + "i" 2sqrt(3)`
Write in polar form of the following complex numbers
`3 - "i"sqrt(3)`
Write in polar form of the following complex numbers
– 2 – i2
Write in polar form of the following complex numbers
`("i" - 1)/(cos pi/3 + "i" sin pi/3)`
Find the rectangular form of the complex numbers
`(cos pi/6 "i" sin pi/6)(cos pi/12 + "i" sin pi/12)`
Find the rectangular form of the complex numbers
`(cos pi/6 - "i" sin pi/6)/(2(cos pi/3 + "i" sin pi/3))`
If (x1 + iy1)(x2 + iy2)(x3 + iy3) ... (xn + iyn) = a + ib, show that `(x_1^2 + y_1^2)(x_2^2 + y_2^2)(x_3^2 + y_3^2) ... (x_"n"^2 + y_"n"^2)` = a2 + b2
If (x1 + iy1)(x2 + iy2)(x3 + iy3) ... (xn + iyn) = a + ib, show that `sum_("r" = 1)^"n" tan^-1 (y_"r"/x_"r") = tan^-1 ("b"/"a") + 2"k"pi, "k" ∈ "z"`
If `(1 + z)/(1 - z)` = cos 2θ + i sin 2θ, show that z = i tan θ
If cos α + cos β + cos γ = sin α + sin β + sin γ = 0, show that cos 3α + cos 3β + cos 3γ = 3 cos (α + β + γ)
If cos α + cos β + cos γ = sin α + sin β + sin γ = 0. then show that sin 3α + sin 3β + sin 3γ = 3 sin(α + β + γ)
Choose the correct alternative:
If z is a complex number such that z ∈ C\R and `"z" + 1/"z"` ∈ R, then |z| is
Choose the correct alternative:
If `(z - 1)/(z + 1)` purely imaginary then |z| is
Choose the correct alternative:
The principal argument of `3/(-1 + "i")` is
Choose the correct alternative:
The principal argument of (sin 40° + i cos 40°)5 is
