1
Question
If 2z1−3z2−z3=0, then z1,z2 and z3 are represented by :
If 2z1−3z2−z3=0, then z1,z2 and z3 are represented by :
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Solution
The correct option is C Three collinear points
Let
z1=x1+iy1
z2=x2+iy2
z3=x3+iy3
Area of Δ formed by z1,z2 &z3
=12∣∣
∣
∣∣x1y11x2y21x3y31∣∣
∣
∣∣
R1=R1−R2
Area =12∣∣
∣
∣∣x1−x2y1−y20x2y21x3y31∣∣
∣
∣∣
R3=R3−R2
Area=12∣∣
∣
∣∣x1−x2y1−y20x2y21x3−x2y3−y20∣∣
∣
∣∣⟶(i)
2z1−3z2+z3=0
2z1−2z2−z2+z3=0
z3−z2=2(z2−z1)∴x3−x2=2(x2−x1) ; y3−y2=2(y2−y1)
Area=12∣∣
∣
∣∣x1−x2y1−y20x2y21−2(x1−x2)−2(y1−y2)0∣∣
∣
∣∣
Area=−22∣∣
∣
∣∣x1−x2y1−y20x2y21x1−x2y1−y20∣∣
∣
∣∣=0(∵RowsR1&R3areidentical)
Since Area=0 ∴z1,z2&z3 are collinear
Let
z1=x1+iy1
z2=x2+iy2
z3=x3+iy3
Area of Δ formed by z1,z2 &z3
=12∣∣ ∣ ∣∣x1y11x2y21x3y31∣∣ ∣ ∣∣
R1=R1−R2
Area =12∣∣ ∣ ∣∣x1−x2y1−y20x2y21x3y31∣∣ ∣ ∣∣
R3=R3−R2
Area=12∣∣ ∣ ∣∣x1−x2y1−y20x2y21x3−x2y3−y20∣∣ ∣ ∣∣⟶(i)
2z1−3z2+z3=0
2z1−2z2−z2+z3=0
z3−z2=2(z2−z1)∴x3−x2=2(x2−x1) ; y3−y2=2(y2−y1)
Area=12∣∣ ∣ ∣∣x1−x2y1−y20x2y21−2(x1−x2)−2(y1−y2)0∣∣ ∣ ∣∣
Area=−22∣∣ ∣ ∣∣x1−x2y1−y20x2y21x1−x2y1−y20∣∣ ∣ ∣∣=0(∵RowsR1&R3areidentical)
Since Area=0 ∴z1,z2&z3 are collinear
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