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If |z + i| = |z - i|, prove that z is re...

If |z + i| = |z - i|, prove that z is real.

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Let z = (x + iy). Then, `|z+i)^(2) = |z - i|^(2)`
`rArr" "|(x+iy)+i|^(2) = |(x+iy)-i|^(2)`
`rArr" "|x+(y+1)i|^(2)=|x+(y-1)i|^(2) rArr x^(2) + (y+1)^(2) = x^(2) + (y-1)^(2)`
`rArr" "(y + 1)^(2) - (y - 1)^(2) = 0 rArr 4y = 0 rArr y = 0`.
Hence, z is purely real.
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RS AGGARWAL-COMPLEX NUMBERS AND QUADRATIC EQUATIONS-EXERCISE 5 B
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  2. Find the real values of theta for which the complex number (1+i costhe...

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  3. If |z + i| = |z - i|, prove that z is real.

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  4. Give an example of two complex numbers z(1) and z(2) such that z(1) ne...

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  5. Find the conjugate of each of the following : {:((i),(-5-2i),(ii),(1...

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  6. Find the modulus of each of the following : {:((i),(3+sqrt(-5)),(ii)...

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  7. Find the multiplicative inverse of each of the following : (i)" "(1-...

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  8. If ((1-i)/(1+i))^(100)=a+i b ,\ fin d\ (a , b)

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  9. If ((1+i)/(1-i))^3-((1-i)/(1+i))^3=x+i y ,\ fin d\ (x , y)

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  10. If x + i y =(a+i b)/(a-i b),prove that x^2+y^2=1.

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  11. If a+i b=(c+i)/(c-i) , where c is real, prove that:a^2+b^2=1a n d b/a=...

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  12. Show that (1-i)^(n)(1-(1)/(i))^(n)=2^(n) for all n in N

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  13. What is the smallest positive integer n for which (1+i)^(2n)=(1-i)^(2n...

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  14. Prove that: x^4=4=(x+1+i)(x+1-i)(x-1+i(x-1-i)dot)

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  15. if a = costheta +i sin theta , prove that (1+a)/(1-a) = cot(theta/2)i

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  16. If z1=2-i ,\ z2=1+i , find |(z1+z2+1)/(z1-z2+i)|

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  17. Find the real values of x and y for which : {:((i),(1-i)x+(1+i)y=1 -...

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  18. Find the real numbers x\ a n d\ y , if (x-i y)(3+5i) is the conjugate ...

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  19. Find real values of xa n dy for which the complex numbers -3+i x^2ya n...

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  20. If z=2-3i show that z^2=4z+13=0 and hence find the value of 4z^3-3z^2+...

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