Given that arg((z-1)/(z-2i)) = pi/4 describe the locus of P and sketch this on an argand diagram. Show that the point (1,3) lies on this locus.
I know it is an arc subtended from the points (0,2) and (1,0) but I do not know how to find the centre of the circle which I am guessing is how I would go on to draw the diagram accurately and answer the second part about the point (1,3)
To find the locus of points P for the given equation, we first simplify the expression (z-1)/(z-2i):
Let z = x + yi, where x and y are real numbers.
Substituting z into the expression, we have:
[(x + yi) - 1]/[(x + yi) - 2i]
Simplifying this expression:
(x + yi - 1)/(x + yi - 2i)
To find the argument of this expression, we can take the argument as the angle of the complex number in the form r(cosθ + isinθ), where r is the magnitude and θ is the argument.
Taking the argument of the numerator (x + yi - 1):
arg(x + yi - 1) = θ₁
Taking the argument of the denominator (x + yi - 2i):
arg(x + yi - 2i) = θ₂
Since the argument of a quotient is the difference of the arguments of the numerator and denominator, we have:
arg((z-1)/(z-2i)) = θ₁ - θ₂
Given that arg((z-1)/(z-2i)) = π/4, we can write:
π/4 = θ₁ - θ₂
Now, let's find the locus of P by substituting points (x, y) into the expression and solving for θ:
Case 1: Let P = (x, y) be the point (0, 2):
Substituting (x, y) = (0, 2) into (x + yi - 1)/(x + yi - 2i):
[(0 + 2i - 1)/((0 + 2i) - 2i)] = [-1/(2i - 2i)] = [-1/0] = undefined
Since this point is undefined, it does not lie on the locus.
Case 2: Let P = (x, y) be the point (1, 0):
Substituting (x, y) = (1, 0) into (x + yi - 1)/(x + yi - 2i):
[(1 + 0i - 1)/((1 + 0i) - 2i)] = [0/(1 - 2i)] = 0
Since the result is 0, the argument θ of this point is 0. Therefore, the point (1, 0) lies on the locus.
Case 3: Let P = (x, y) be the point (1, 3):
Substituting (x, y) = (1, 3) into (x + yi - 1)/(x + yi - 2i):
[(1 + 3i - 1)/((1 + 3i) - 2i)] = [3i/(1 + i)] = [(3i(1 - i))/((1 + i)(1 - i))] = [(3i - 3)/2] = (3/2)i - 3/2
To find the argument θ of this point, we need to find its equivalent in the form r(cosθ + isinθ):
(3/2)i - 3/2 = √[(3/2)^2 + (-3/2)^2] * cosθ + i√[(3/2)^2 + (-3/2)^2] * sinθ
Simplifying the expressions:
[(3/2)i - 3/2] = (√9/4) * [cosθ + isinθ]
(3i - 3)/2 = (3/2) * [cosθ + isinθ]
Comparing the coefficients:
3i - 3 = 3(cosθ + isinθ)
Since the real parts are equal and the imaginary parts are equal, we can equate them individually:
3i = 3isinθ
-3 = 3cosθ
Therefore, sinθ = 1 and cosθ = -1.
From trigonometry, we know that sinθ = 1 when θ = π/2, and cosθ = -1 when θ = π.
Hence, the argument θ of the point (1, 3) is either π/2 or π.
To summarize, the locus of points P for the equation arg((z-1)/(z-2i)) = π/4 is an arc subtended from the points (0, 2) and (1, 0) on the Argand diagram. The point (1, 3) lies on this locus since its argument can be either π/2 or π.
To accurately sketch this on an Argand diagram, you can plot the points (0, 2), (1, 0), and (1, 3) and draw an arc passing through these points. The center of the circle can be found by drawing a perpendicular bisector of the line segment joining (0, 2) and (1, 0). The point where the perpendicular bisector intersects the line joining (0, 2) and (1, 0) is the center of the circle.