In an Argand diagram O represents the origin, A represents the complex number `2Z^2`  and B represents the complex number `3/z^2` . Does the point representing Z lie on the line passing  through...

In an Argand diagram O represents the origin, A represents the complex number `2Z^2`  and B represents the complex number `3/z^2` .

Does the point representing Z lie on the line passing  through O and B?Justify your answer.

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should use the following notation for the complex number z, such that:

`z = x + i*y => z^2 = (x + i*y)^2 `

Expanding the square yields:

`z^2 = x^2 + 2i*x*y + i^2*y^2`

Since `i^2 = -1` yields:

`z^2 = x^2 - y^2 + 2i*x*y `

The point B is given by the following complex number `3/z^2` , hence, you should come up with the following notation, such that:

`z_1 = 3/z^2 => z_1 = 3/(x^2 - y^2 + 2i*x*y)`

Performing the multiplication by its conjugate, yields:

`z_1 = 3(x^2 - y^2 - 2i*x*y)/((x^2 - y^2)^2 + 4xy)`

The point B is represented by the coordinates `(3(x^2 - y^2)/((x^2 - y^2)^2 + 4xy),- (6*x*y)/((x^2 - y^2)^2 + 4xy))`

The line OB is given by the following equation, such that:

`(x_B - x_O)/(x - x_O) = (y_B - y_O)/(y - y_O)`

`- (6*x*y)/((x^2 - y^2)^2 + 4xy)*x = y*3(x^2 - y^2)/((x^2 - y^2)^2 + 4xy)`

The point z represented by the coordinates (x,y) belongs to the line OB if replacing its coordinates in the equation of the line OB, the equation holds.

`-6x^2y = 3y(x^2 - y^2) => -2x^2 = x^2 - y^2 => 3x^2 = y^2`

Since replacing the coordinates of the point z in equation of the line OB the equation does not hold yields that z does not belong to OB.

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