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Question

If P is the point in the argand diagram corresponding to the complex number 3+i and if OPQ is an isosceles right angled triangle, right angled at O, then Q represents the complex number


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Solution

Step 1: Finding the value slopes

For the graphical representation of complex numbers, an argand diagram is used, if the complex number is in the form of x+iy, in the complex plane. Similar to the x−axis and y−axis in the two-dimensional geometry.

Given the complex number P=3+i, and Ois center

Let us draw a figure, according to the question for understanding

The figure shows an Isosceles right-angled triangle OPQ

If two lines are perpendicular to each other then the product of their slopes will be equal to -1, that is m1×m2=-1

The slope of line OP is

m1=1030=13

Let Q=x,y , then the slope m2 of line OQ;

m2=y0x0=yx

Step 2: Substitute the values of slopes into m1×m2=-1, we get

13×yx=-1y=-3x......(i)

Step 3: Using the properties of isosceles triangles, Congruent sides of the isosceles triangle are equal, then OP=OQ,

Use the distance formula between two points d=(x2x1)2+(y2y1)2, then

OP=(30)2+(10)2=3+1=4

Similarly,

OQ=(x0)2+(y0)2=x2+y2

Since OP=OQ, then

OP=OQOP2=OQ24=x2+y2

Put y=3x from equation (iii), in the above equation,

4=x2+(3x)24=x2+3x24=4x2x2=1x=±1

Step 4: Substitute x=1andx=1 in the equation (iii) to find the ordered pair Qx,y

When x=1, then y=3, and x=-1, then y=3,

Thus, there can be two possible values of Qx,y are 1,-3 and -1,3.

Hence, the answer can be written Q represents the complex number either 1-i3 or -1+i3.


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