Complex Loci with Arguments
I need to find the locus of points (on an Argand diagram) such that:
(i) $arg(z-(-1-4i)) + arg(z-(5+8i)) =0$
(ii) $arg(z-(-1-4i)) + arg(z-(5+8i)) = pi/2$
I could not see a way to solve these problems other than plotting arbitrary points and trying to observe a general pattern.
I am aware that $arg(z-(-1-4i)) - arg(z-(5+8i)) = pi/2$ is a semicircle, and for other angles, say $pi/3$ or $pi/4 $, part of the arc of a circle, but this is only because I knew that this equation represents the locus of points that made a certain angle between the two complex numbers. I was unable to find a similar representation, however, for (i) and (ii).
I am also interested in whether problems (i) and (ii) can be generalised to any angle between $0$ to $pi$. Any help here would be greatly appreciated.
complex-numbers locus
asked Oct 6 at 19:47
NKRsolutions
588
add a comment |
I need to find the locus of points (on an Argand diagram) such that:
(i) $arg(z-(-1-4i)) + arg(z-(5+8i)) =0$
(ii) $arg(z-(-1-4i)) + arg(z-(5+8i)) = pi/2$
I could not see a way to solve these problems other than plotting arbitrary points and trying to observe a general pattern.
I am aware that $arg(z-(-1-4i)) - arg(z-(5+8i)) = pi/2$ is a semicircle, and for other angles, say $pi/3$ or $pi/4 $, part of the arc of a circle, but this is only because I knew that this equation represents the locus of points that made a certain angle between the two complex numbers. I was unable to find a similar representation, however, for (i) and (ii).
I am also interested in whether problems (i) and (ii) can be generalised to any angle between $0$ to $pi$. Any help here would be greatly appreciated.
complex-numbers locus
asked Oct 6 at 19:47
NKRsolutions
588
add a comment |
I need to find the locus of points (on an Argand diagram) such that:
(i) $arg(z-(-1-4i)) + arg(z-(5+8i)) =0$
(ii) $arg(z-(-1-4i)) + arg(z-(5+8i)) = pi/2$
I could not see a way to solve these problems other than plotting arbitrary points and trying to observe a general pattern.
I am aware that $arg(z-(-1-4i)) - arg(z-(5+8i)) = pi/2$ is a semicircle, and for other angles, say $pi/3$ or $pi/4 $, part of the arc of a circle, but this is only because I knew that this equation represents the locus of points that made a certain angle between the two complex numbers. I was unable to find a similar representation, however, for (i) and (ii).
I am also interested in whether problems (i) and (ii) can be generalised to any angle between $0$ to $pi$. Any help here would be greatly appreciated.
complex-numbers locus
asked Oct 6 at 19:47
NKRsolutions
588
I need to find the locus of points (on an Argand diagram) such that:
(i) $arg(z-(-1-4i)) + arg(z-(5+8i)) =0$
(ii) $arg(z-(-1-4i)) + arg(z-(5+8i)) = pi/2$
I could not see a way to solve these problems other than plotting arbitrary points and trying to observe a general pattern.
I am aware that $arg(z-(-1-4i)) - arg(z-(5+8i)) = pi/2$ is a semicircle, and for other angles, say $pi/3$ or $pi/4 $, part of the arc of a circle, but this is only because I knew that this equation represents the locus of points that made a certain angle between the two complex numbers. I was unable to find a similar representation, however, for (i) and (ii).
I am also interested in whether problems (i) and (ii) can be generalised to any angle between $0$ to $pi$. Any help here would be greatly appreciated.
complex-numbers locus
complex-numbers locus
asked Oct 6 at 19:47
NKRsolutions
588
asked Oct 6 at 19:47
NKRsolutions
588
edited Oct 7 at 6:53
asked Oct 6 at 19:47
NKRsolutions
588
asked Oct 6 at 19:47
NKRsolutions
588
asked Oct 6 at 19:47
NKRsolutions
588
588
add a comment |
add a comment |
1 Answer
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Denote the points $A(-1-4i),; B(5+8i),; M(z).$
(i) is equivalent to $$arg(z-(-1-4i)) =- arg(z-(5+8i))$$ This signifies that the direction from $A$ towards $M$ is opposite to the one from $B$ towards $M.$ The locus of points $M(z)$ is the segment $AB$ except $A,B.$
answered Nov 26 at 21:51
user376343
2,7882822
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Denote the points $A(-1-4i),; B(5+8i),; M(z).$
(i) is equivalent to $$arg(z-(-1-4i)) =- arg(z-(5+8i))$$ This signifies that the direction from $A$ towards $M$ is opposite to the one from $B$ towards $M.$ The locus of points $M(z)$ is the segment $AB$ except $A,B.$
answered Nov 26 at 21:51
user376343
2,7882822
add a comment |
Denote the points $A(-1-4i),; B(5+8i),; M(z).$
(i) is equivalent to $$arg(z-(-1-4i)) =- arg(z-(5+8i))$$ This signifies that the direction from $A$ towards $M$ is opposite to the one from $B$ towards $M.$ The locus of points $M(z)$ is the segment $AB$ except $A,B.$
answered Nov 26 at 21:51
user376343
2,7882822
add a comment |
Denote the points $A(-1-4i),; B(5+8i),; M(z).$
(i) is equivalent to $$arg(z-(-1-4i)) =- arg(z-(5+8i))$$ This signifies that the direction from $A$ towards $M$ is opposite to the one from $B$ towards $M.$ The locus of points $M(z)$ is the segment $AB$ except $A,B.$
answered Nov 26 at 21:51
user376343
2,7882822
Denote the points $A(-1-4i),; B(5+8i),; M(z).$
(i) is equivalent to $$arg(z-(-1-4i)) =- arg(z-(5+8i))$$ This signifies that the direction from $A$ towards $M$ is opposite to the one from $B$ towards $M.$ The locus of points $M(z)$ is the segment $AB$ except $A,B.$
answered Nov 26 at 21:51
user376343
2,7882822
answered Nov 26 at 21:51
user376343
2,7882822
answered Nov 26 at 21:51
user376343
2,7882822
answered Nov 26 at 21:51
user376343
2,7882822
2,7882822
add a comment |
add a comment |
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