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If z=2 + t + i sqrt(3-t^2), where t is ...

If `z=2 + t + i sqrt(3-t^2)`, where t is real and `t^2 lt 3`, show that the modulus of `(z+1)/(z-1)` is independent of t. Also show that the locus of the points z for different values of t is a circle and finds its centre and radius.

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A DAS GUPTA-COMPLEX NUMBERS-EXERCISE
  1. If the points P and Q represent the complex numbers z and iz then angl...

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  2. If z1ne-z2 and |z1+z2|=|1/z1 + 1/z2| then :

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  3. Find z, if |(z+1)/(z+i)|=1

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  4. If in the Argand plane z1,z2,z3 and z4 are four points such that |z1|=...

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  5. State true or false: If zne0 then argz+argbarz=0.

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  6. Let z1 and z2 be the roots of z^2+pz+q=0. Then the points represented ...

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  7. If z1,z2 are nonzero complex numbers then |(z1)/(|z1|)+(z2)/(|z2|)|le2...

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  8. The nth roots of -1 can be n terms of a GP.

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  9. In the following, each question has one or more than one correct answe...

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  10. If |z-3i|=3 and amp z in (0, pi/2), then cot(amp z)-theta/z is eqaul ...

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  11. If zne0 then amp z +ampbarz = .

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  12. x+iy=((sqrt3+i)/(1-sqrt3i))^25 then x=, y=.

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  13. If alpha1,alpha2,alpha3,alpha4 be the roots of x^5-1=0 then prove that...

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  14. Consider the quadratic equation az^(2)+bz+c=0 where a,b,c and non-zero...

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  15. If |z^2-4|=2|z| then find the greatest value of |z|.

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  16. If z=2 + t + i sqrt(3-t^2), where t is real and t^2 lt 3, show that t...

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  17. If |z1|=1,|z2|=1 then prove that |z1+z2|^2+|z1-z2|^2 =4.

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  18. z1,z2 are represented by two consecutive vertices of a rhombus, the an...

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  19. Prove that t^(2) + 3t + 3 is a factor of ( t+1)^(n+1)^(n+1) + (t+2)^(2...

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  20. If x^2-x+1 =0 then find the value of (x+1/x)^2+(x^2+1/x^2)^2+(x^3+1/x^...

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