1
Question
If z1,z2,z3 are three collinear points in arranged plane , then ∣∣
∣∣z1¯z11z2¯z21z3¯z31∣∣
∣∣ =
If z1,z2,z3 are three collinear points in arranged plane , then ∣∣
∣∣z1¯z11z2¯z21z3¯z31∣∣
∣∣ =
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Solution
The correct option is A 0
If z1,z2z3 are collinear.⎡⎢⎣x1y1z1x2y2z2111⎤⎥⎦=0⎧⎨⎩z1=x1+iy1z2=x2+iy2z3=x3+iy3
⎡⎢⎣z1¯z11z2¯z21z3¯z31⎤⎥⎦=⎡⎢⎣x1+iy1x1−iy11x2+iy2x2−iy21x3+iy3x3−iy31⎤⎥⎦=⎡⎢⎣x1x1−iy11x2x2−iy21x3x3−iy31⎤⎥⎦+⎡⎢⎣iy1x1−iy11iy2x2−iy21iy3x3−iy31⎤⎥⎦=⎡⎢⎣x1x11x2x21x3x31⎤⎥⎦+⎡⎢⎣x1−iy11x2−iy21x3−iy31⎤⎥⎦+⎡⎢⎣iy1x11iy2x21iy3x31⎤⎥⎦+⎡⎢⎣iy1−iy11iy2−iy21iy3−iy31⎤⎥⎦=0−⎡⎢⎣x1iy11x2iy21x3iy31⎤⎥⎦+⎡⎢⎣iy1x11iy2x21iy3x31⎤⎥⎦+0
=0⎛⎜⎝∵⎡⎢⎣x1iy11x2iy21x3iy31⎤⎥⎦=0⎞⎟⎠
So A option is correct.
If z1,z2z3 are collinear.
⎡⎢⎣x1y1z1x2y2z2111⎤⎥⎦=0⎧⎨⎩z1=x1+iy1z2=x2+iy2z3=x3+iy3
⎡⎢⎣z1¯z11z2¯z21z3¯z31⎤⎥⎦=⎡⎢⎣x1+iy1x1−iy11x2+iy2x2−iy21x3+iy3x3−iy31⎤⎥⎦
=⎡⎢⎣x1x1−iy11x2x2−iy21x3x3−iy31⎤⎥⎦+⎡⎢⎣iy1x1−iy11iy2x2−iy21iy3x3−iy31⎤⎥⎦
=⎡⎢⎣x1x11x2x21x3x31⎤⎥⎦+⎡⎢⎣x1−iy11x2−iy21x3−iy31⎤⎥⎦+⎡⎢⎣iy1x11iy2x21iy3x31⎤⎥⎦+⎡⎢⎣iy1−iy11iy2−iy21iy3−iy31⎤⎥⎦
=0−⎡⎢⎣x1iy11x2iy21x3iy31⎤⎥⎦+⎡⎢⎣iy1x11iy2x21iy3x31⎤⎥⎦+0
=0⎛⎜⎝∵⎡⎢⎣x1iy11x2iy21x3iy31⎤⎥⎦=0⎞⎟⎠
So A option is correct.
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