## anonymous 5 years ago |z+4| + |Z-4| = 10 is contained or equal to a. ellipse with eccentricity 4/5 b. hyperbola with eccentricity 4/5 c. parabola d. none of these. Where Z is complex Number ????? Pls Help How to Solve These Types Of Question And what are complex number conditions for ellipse, hyperbola, circle, parabola etc.....

1. anonymous

please anyone, how to solve this ????

2. anonymous

For a circle centered at $z_0=a+bi$ with radius C the condition must be $\left|z-z_0\right|=C^2$

3. anonymous

what is value of Z here ...... its ans is " option a" but how to solve this ??

4. anonymous

The value of Z is an arbitrary complex number... you need an infinite number of z who satisfy the condition to draw the circle on the complex plain...

5. anonymous

This is an ellipse with equation$\frac{x^2}{5^2}+\frac{y^2}{3^3}=1$which implies eccentricity of$e=\sqrt{1-(\frac{3}{5})^2}=\frac{4}{5}$

6. anonymous

$z=x+iy \rightarrow |z|=\sqrt{x^2+y^2}$this is what you use to solve this.

7. anonymous

$|z+4|=|(x+iy)+4|=|(x+4)+iy|=\sqrt{(x+4)^2+y^2}$Similarly with the second magnitude.

8. anonymous

by definition, ellipse is the locus of all those points the sum of whose distance from two fixed point is a constant here either u can solve the eq by putting z= x+iy or u can directly find it

9. anonymous

10. anonymous

|z+4| is the distance of all z from (-4,0) and |z-4| is the distance of all z from (4,0)

11. anonymous

as the sum of distances from two fixed points is 10, a constant, thus it is an ellipse

12. anonymous

thanks Lokisan,conighion, uzma , and what in case of hyperbola .............. lokisan very very thanks, awesome help by u and others

13. anonymous

in case of hyperbola, we take the diffence from two fixed points

14. anonymous

difference*

15. anonymous

ok

16. anonymous

you're welcome

17. anonymous

From: http://en.wikipedia.org/wiki/Conic_section "Over the complex numbers ellipses and hyperbolas are not distinct"

18. anonymous

Hyperbolas typically have |f(z)| - |g(z)| = a but you have to check using definitions for z and absolute value, since sometimes you can get degenerate conics from these equations (like straight lines).

19. anonymous

so the basic idea is to solve the complex equation and fid the equation and then compare it with the equation of circle ellipse, hyper etc

20. anonymous

Pretty much. Your finding the locus of points that satisfy the condition you're given.

21. anonymous

k,this question is asked every year in Many Indian Engineering Entrance Exam in different form , thanks all, now i get it, this help me lot.

22. anonymous

np probs. good luck with it :)