Advertisements
Advertisements
प्रश्न
If arg(z – 1) = arg(z + 3i), then find x – 1 : y. where z = x + iy.
उत्तर
Given that: arg(z – 1) = arg(z + 3i)
⇒ arg[x + yi – 1] = arg[x + yi + 3i]
⇒ arg[(x – 1) + yi] = arg[x + (y + 3)i]
⇒ `tan^-1 y/(x - 1) = tan^-1 (y + 3)/x`
⇒ `y/(x - 1) = (y + 3)/x`
⇒ xy = (x – 1)(y + 3)
⇒ xy = xy + 3x – y – 3
⇒ 3x – y = 3
⇒ 3x – 3 = y
⇒ 3(x – 1) = y
⇒ `((x - 1))/y = 1/3`
⇒ x – 1 : y = 1 : 3
Hence, x – 1 : y = 1 : 3.
APPEARS IN
संबंधित प्रश्न
Find the modulus and the argument of the complex number `z = – 1 – isqrt3`
Convert the given complex number in polar form: 1 – i
Convert the given complex number in polar form: – 1 + i
Convert the given complex number in polar form: – 1 – i
Convert the given complex number in polar form: –3
Convert the given complex number in polar form `sqrt3 + i`
Convert the following in the polar form:
`(1+7i)/(2-i)^2`
Convert the following in the polar form:
`(1+3i)/(1-2i)`
If the imaginary part of `(2z + 1)/(iz + 1)` is –2, then show that the locus of the point representing z in the argand plane is a straight line.
Let z1 and z2 be two complex numbers such that `barz_1 + ibarz_2` = 0 and arg(z1 z2) = π. Then find arg (z1).
Let z1 and z2 be two complex numbers such that |z1 + z2| = |z1| + |z2|. Then show that arg(z1) – arg(z2) = 0.
If |z| = 2 and arg(z) = `pi/4`, then z = ______.
The locus of z satisfying arg(z) = `pi/3` is ______.
What is the polar form of the complex number (i25)3?
Show that the complex number z, satisfying the condition arg`((z - 1)/(z + 1)) = pi/4` lies on a circle.
If for complex numbers z1 and z2, arg (z1) – arg (z2) = 0, then show that `|z_1 - z_2| = |z_1| - |z_2|`.
Write the complex number z = `(1 - i)/(cos pi/3 + i sin pi/3)` in polar form.
If z and w are two complex numbers such that |zw| = 1 and arg(z) – arg(w) = `pi/2`, then show that `barz`w = –i.
If |z| = 4 and arg(z) = `(5pi)/6`, then z = ______.
State True or False for the following:
Let z1 and z2 be two complex numbers such that |z1 + z2| = |z1| + |z2|, then arg(z1 – z2) = 0.
The value of arg (x) when x < 0 is ______.
If arg(z) < 0, then arg(–z) – arg(z) = ______.
