For complex numbers z satisfying arg (z - u) = pi/4, find the least possible value of IzI
where u is -3i
How is it solved ???
where u is -3i
How is it solved ???
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The complex numbers w satisfying arg(w) = pi/4 are the numbers e^(i pi/4) * t, with t > 0.
If z - u has this form then z - u = e^(i pi/4) t for some t > 0, and hence
z = e^(i pi/4) t + u
Writing e^(i pi/4) = cos(pi/4) + i sin(pi/4) = 1/sqrt(2) + i/sqrt(2) and using u = -3i we see that arg(z - u) = pi/4 if and only if there is a real number t > 0 with
z = t/sqrt(2) + i [t/sqrt(2) - 3].
For such z we have
|z|^2 = (t/sqrt(2))^2 + (t/sqrt(2) - 3)^2 = (1/2) t^2 + (1/2) t^2 - (6/sqrt(2)) t + 9
= t^2 - 3 sqrt(2) t + 9.
Using algebra (writing t^2 - 3 sqrt(2) t + 9 as (t - (3/2) sqrt(2))^2 + 9 - [(3/2) sqrt(2)]^2) or doing calculus here we see that the value of t > 0 that makes this expression a minimum is t = (3/2) sqrt(2). The corresponding value of |z|^2 is 9 - ((3/2) sqrt(2))^2 = 9 - (9/4) * 2 = 9 - 9/2 = 9/2. This is the least possible value of |z|^2 for z satisfying arg(z - u) = pi/4. The least possible value of |z| is then of course sqrt(9/2).
You can also think of this as a geometric problem. A complex number z satisfies arg(z - u) = pi/4 if and only if the oriented line segment drawn from z to u points in a direction that makes a 45 degree angle with the direction of the positive x-axis. If you visualize this set of z geometrically you see it is the set of points z = x + iy, where (x,y) lies on the line through (0,3) with slope 1 and also where x < 0. Geometrically this is a ray (the set of all complex numbers of the form x + (x + 3) i, with x < 0) and you are asked for the distance from (0,0) to the closest point on this ray. Drawing a picture you can see that the closest point on the ray is the midpoint of the line segment with endpoints (-3,0) and (0,3), namely the point (-3/2, 3/2) so the minimum distance is sqrt((3/2)^2 + (3/2)^2) = sqrt(9/2).
If z - u has this form then z - u = e^(i pi/4) t for some t > 0, and hence
z = e^(i pi/4) t + u
Writing e^(i pi/4) = cos(pi/4) + i sin(pi/4) = 1/sqrt(2) + i/sqrt(2) and using u = -3i we see that arg(z - u) = pi/4 if and only if there is a real number t > 0 with
z = t/sqrt(2) + i [t/sqrt(2) - 3].
For such z we have
|z|^2 = (t/sqrt(2))^2 + (t/sqrt(2) - 3)^2 = (1/2) t^2 + (1/2) t^2 - (6/sqrt(2)) t + 9
= t^2 - 3 sqrt(2) t + 9.
Using algebra (writing t^2 - 3 sqrt(2) t + 9 as (t - (3/2) sqrt(2))^2 + 9 - [(3/2) sqrt(2)]^2) or doing calculus here we see that the value of t > 0 that makes this expression a minimum is t = (3/2) sqrt(2). The corresponding value of |z|^2 is 9 - ((3/2) sqrt(2))^2 = 9 - (9/4) * 2 = 9 - 9/2 = 9/2. This is the least possible value of |z|^2 for z satisfying arg(z - u) = pi/4. The least possible value of |z| is then of course sqrt(9/2).
You can also think of this as a geometric problem. A complex number z satisfies arg(z - u) = pi/4 if and only if the oriented line segment drawn from z to u points in a direction that makes a 45 degree angle with the direction of the positive x-axis. If you visualize this set of z geometrically you see it is the set of points z = x + iy, where (x,y) lies on the line through (0,3) with slope 1 and also where x < 0. Geometrically this is a ray (the set of all complex numbers of the form x + (x + 3) i, with x < 0) and you are asked for the distance from (0,0) to the closest point on this ray. Drawing a picture you can see that the closest point on the ray is the midpoint of the line segment with endpoints (-3,0) and (0,3), namely the point (-3/2, 3/2) so the minimum distance is sqrt((3/2)^2 + (3/2)^2) = sqrt(9/2).