Advertisements
Advertisements
प्रश्न
Find the equation in cartesian coordinates of the locus of z if |z – 2 – 2i| = |z + 2 + 2i|
Advertisements
उत्तर
Let z = x + iy
|z – 2 – 2i| = |z + 2 + 2i|
∴ |x + iy – 2 – 2i| = |x + iy + 2 + 2i|
∴ |(x – 2) + i(y – 2)| = |(x + 2) + i(y + 2)|
∴ `sqrt((x - 2)^2 + (y - 2)^2) = sqrt((x + 2)^2 + (y + 2)^2)`
∴ (x – 2)2 + (y – 2)2 = (x + 2)2 + (y + 2)2
∴ x2 – 4x + 4 + y2 – 4y + 4 = x2 + 4x + 4 + y2 + 4y + 4
∴ –4x – 4y = 4x + 4y
∴ 8x + 8y = 0
∴ x + y = 0
∴ y = −x
APPEARS IN
संबंधित प्रश्न
If ω is a complex cube root of unity, show that `((a + bomega + comega^2))/("c" + aomega + bomega^2) = omega^2`.
If α and β are the complex cube roots of unity, show that α2 + β2 + αβ = 0.
If ω is a complex cube root of unity, then prove the following: (a + b) + (aω + bω2) + (aω2 + bω) = 0.
Find the value of ω21
Find the value of ω–30
If ω is a complex cube root of unity, show that (2 − ω)(2 − ω2) = 7
If ω is a complex cube root of unity, show that (1 + ω − ω2)6 = 64
If ω is a complex cube root of unity, show that (1 + ω)3 − (1 + ω2)3 = 0
If ω is a complex cube root of unity, show that (3 + 3ω + 5ω2)6 − (2 + 6ω + 2ω2)3 = 0
If ω is a complex cube root of unity, show that `("a" + "b"ω + "c"ω^2)/("c" + "a"ω + "b"ω^2)` = ω2
If ω is a complex cube root of unity, show that (a − b) (a − bω) (a − bω2) = a3 − b3
If ω is a complex cube root of unity, find the value of (1 + ω2)3
If ω is a complex cube root of unity, find the value of (1 − ω − ω2)3 + (1 − ω + ω2)3
If α and β are the complex cube root of unity, show that α2 + β2 + αβ = 0
If α and β are the complex cube root of unity, show that α4 + β4 + α−1β−1 = 0
Find the equation in cartesian coordinates of the locus of z if |z| = 10
Find the equation in cartesian coordinates of the locus of z if |z − 5 + 6i| = 5
Find the equation in cartesian coordinates of the locus of z if |z + 8| = |z – 4|
Select the correct answer from the given alternatives:
If ω is a complex cube root of unity, then the value of ω99+ ω100 + ω101 is :
Answer the following:
If ω is a complex cube root of unity, prove that (1 − ω + ω2)6 +(1 + ω − ω2)6 = 128
If 1, α1, α2, ...... αn–1 are the roots of unity, then (1 + α1)(1 + α2) ...... (1 + αn–1) is equal to (when n is even) ______.
If the cube roots of the unity are 1, ω and ω2, then the roots of the equation (x – 1)3 + 8 = 0, are ______.
Simplify the following and express in the form a + ib.
`(3i^5 + 2i^7 + i^9)/(i^6 + 2i^8 + 3i^18)`
If w is a complex cube root of unity, show that
`((a + bw + cw^2)) /( c + aw + bw^2 )= w^2`
If w is a complex cube root of unity, show that `((a+bw+cw^2))/(c+aw+bw^2) =w^2`
If ω is a complex cube root of unity, then prove the following.
(ω2 + ω −1)3 = −8
If ω is a complex cube-root of unity, then prove the following:
(a + b) + (aω + bω2) + (aω2 + bω) = 0
If w is a complex cube root of unity, show that `((a + bω + cω^2))/(c + aω + bω^2) = ω^2`
If w is a complex cube-root of unity, then prove the following
(w2 + w - 1)3 = - 8
If w is a complex cube root of unity, show that `((a+bw+cw^2))/(c+aw+bw^2)=w^2`
If w is a complex cube root of unity, show that `((a + bomega + comega^2))/(c + aomega + bomega^2) = w^2`
If ω is a complex cube root of unity, show that `((a + b\omega + c\omega^2))/(c + a\omega + b\omega^2) = \omega^2`
