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 describe the locus of P given arg((z-1)/(z-2i)) = pi/4, we first need to simplify the expression (z-1)/(z-2i) using complex number arithmetic.
Let z = x + yi, where x and y are real numbers.
Then, (z-1) = (x + yi - 1) = (x - 1) + yi
And (z-2i) = (x + yi - 2i) = x + (y - 2)i
Substituting these expressions into the given equation, we have:
arg((z-1)/(z-2i)) = arg(((x - 1) + yi)/(x + (y - 2)i))
To find the argument of a complex number in rectangular form (x + yi), we can use the formula arg(z) = arctan(y/x).
So, arg((z-1)/(z-2i)) = arg(((x - 1) + yi)/(x + (y - 2)i)) = arctan[(y/(x-1))/(y/(y-2))]
Since the argument is given as pi/4, we have:
pi/4 = arctan[(y/(x-1))/(y/(y-2))]
Now, we can simplify this equation and solve for x and y.
Next, let's consider the point (1,3) to check if it lies on this locus.
We substitute x=1 and y=3 into the equation pi/4 = arctan[(y/(x-1))/(y/(y-2))] and check if it holds true.
pi/4 = arctan[(3/(1-1))/(3/(3-2))]
pi/4 = arctan[(3/0)/(3/1)]
pi/4 = arctan[undefined]
Since the expression is undefined, this means that the point (1,3) does not lie on the locus described by arg((z-1)/(z-2i)) = pi/4.
Please note that the locus described by arg((z-1)/(z-2i)) = pi/4 is not a circle but an arc subtended from the points (0,2) and (1,0) on the Argand diagram.
To describe the locus of P and sketch it on an Argand diagram, we need to find the equation of the curve that satisfies the given condition of arg((z-1)/(z-2i)) = pi/4.
Step 1: Write the complex number in its polar form. Let z = x+iy, where x and y are real numbers.
(z-1)/(z-2i) = [(x-1)+iy]/[(x-2i)+iy]
Step 2: Convert the complex number to polar form. The modulus of a complex number in polar form is given by |z| = sqrt(x^2 + y^2), and the argument (arg) is given by θ = arctan(y/x).
The modulus of (z-1)/(z-2i) can be written as |(z-1)/(z-2i)| = |(x-1)+iy|/|x-2i+iy|
The argument (arg) of (z-1)/(z-2i) is given by arg((z-1)/(z-2i)) = arg((x-1)+iy) - arg(x-2i+iy)
Step 3: Find the argument of (z-1)/(z-2i) using the properties of complex numbers.
arg((x-1)+iy) = arctan(y/(x-1))
arg(x-2i+iy) = arctan((x-2)/y)
arg((z-1)/(z-2i)) = arctan(y/(x-1)) - arctan((x-2)/y)
Given that arg((z-1)/(z-2i)) = π/4, we have:
π/4 = arctan(y/(x-1)) - arctan((x-2)/y)
Step 4: Simplify the equation.
Rearranging the equation, we get:
arctan(y/(x-1)) = arctan((x-2)/y) + π/4
Take the tangent of both sides:
tan(arctan(y/(x-1))) = tan(arctan((x-2)/y) + π/4)
y/(x-1) = ((x-2)/y) + 1
Simplifying further,
y^2 = (x-1)(x-2) + y(x-1)
Rearranging and simplifying:
y^2 - y(x-1) = x^2 - 3x + 2
y^2 - xy + x - y = x^2 - 3x + 2
y(y - x + 1) = x^2 - 4x + 2
This equation represents the locus of P.
To sketch this on an Argand diagram, you can follow these steps:
1. Plot the points (0,2) and (1,0) on the Argand diagram.
2. Draw a straight line segment connecting these two points. This segment represents the arc of the locus.
3. To find the center of the circle of which this arc is a part, you need to find the intersection point between the bisector of the segment (0,2) to (1,0) and the perpendicular bisector of this segment.
4. The intersection point that you find will be the center of the circle from which the arc is a part. Draw the circle with this center and the plotted arc.
Now, to show that the point (1,3) lies on this locus, substitute the values of x and y in the equation of the locus:
(1)^2 - (3)(1) + 1 - (3) = 1 - 3 + 1 - 3 = -3
Since the equation is satisfied, it proves that the point (1,3) lies on the locus.