Second part to show is easy. The gradient from P to 1 + 0i is the same as the gradient from 1 + 0i to Q

Edit :

The two tangent values are equal.

The line from P(x,yi) to (1,0i) is parallel to the line from Z-1 (x - 1, yi) to the origin (0,0i). It's a horizontal translation of 1 unit to the left on the real(Re) axis.

So the angle Z-1 makes with the real axis and origin is equal to that made by P with the real axis and (1,0i)

The gradient of the 2 lines is the same
i.e tan [arg(z-1)] = gradient of PQ

Similarly,

The gradient of PQ equals gradient of the line from Q (4/z) to (1,0i).

Reflecting Q (4/z) along the line Re(z) = Im(z) gives -4/z.

Then, horizontal translation of 1 unit to the right on the real axis gives 1 - 4/z, which actually lies on the line from Z - 1 to (0,0i)

This means that 1 - 4/z is a reflection of 4/z along the real axis and tan[arg(1 - 4/z)] is equal to the gradient of line from Z-1 (x - 1, yi) to the origin (0,0i).

i.e tan[arg(1 - 4/z)] = tan [arg(z-1)]



So y/(x - 1) = 4y/(x² + y² - 4x)

y(x² + y² - 4x) = 4y(x - 1)

y(x² + y² - 4x - 4(x - 1)) = 0

y(x² - 8x + 4 + y²) = 0

y(x² - 2(4)(x) + (-4)² + 4 - (-4)² + y²) = 0

y( (x - 4)² + y² - 12) = 0

y = 0 or (x - 4)² + y² = 12

(Shown)

The other way is to directly equate the gradients

(y - 0)/(x - 1) = (-4y/(x² + y²) - 0)/( 4x/(x² + y²) - 1)

y/(x - 1) = -4y/(x² + y²) / (4x - x² - y²)/(x² + y²)

y/(x - 1) = -4y/(4x - x² - y²) = 4y/(x² + y² - 4x)

Solve accordingly to previous comment

great thanks for all helps

Last part :


From earlier parts we can tell that the locus of P is a circle with centre (4,0i) and radius √12 units = 2√3 units

great thanks for all of your hard work，efforts and explanation

The locus of P and Q are points on the same circle , satisfying the equation, with PQ always passing through (1,0i)

So P and Q are always equidistant from the centre since you would form a radius if you draw either P or Q to the centre.


Now the radius from Q (the point 4/z) to the centre (4,0i) can be represented by |4 - 4/z|

i.e distance between two points or length of a line between 2 points.

Since the radius = 2√3 units,

|4 - 4/z| = 2√3

|4(1 - 1/z)| = 2√3

4|1 - 1/z| = 2√3

|1 - 1/z| = ½√3 (shown)

appreciate for all your hard work