If A- B- C are three points in argand plane representing the complex number z1- z2- z3 such that z1- z2-z3-1- where - R- then find the distance of point A from the line joining points B and C-

The correct option is B 0 Let z_1 = x_1+iy_1, z_2 = x_2 +iy_2 and z_3= x_3 +iy_3 z_1 = λ z_2+z_3λ+1 x_1+iy_1 = λ x_2 +λ i y_2 + x_3+i ...

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Standard XII

Mathematics

Image of a Point with Respect to a Line

If A, B, C ar...

Question

If A, B, C are three points in argand plane representing the complex number $z_{1}, z_{2}, z_{3}$ such that $z_{1} = \frac{λ z_{2} + z_{3}}{λ + 1}$ , where $λ ϵ R$ , then find the distance of point A from the line joining points B and C.

0

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λ

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λ + 1

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\frac{λ}{(λ + 1)}

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Solution

The correct option is B $0$
Let $z_{1} = x_{1} + i y_{1}, z_{2} = x_{2} + i y_{2}$ and $z_{3} = x_{3} + i y_{3}$

$z_{1} = \frac{λ z_{2} + z_{3}}{λ + 1}$

$x_{1} + i y_{1} = \frac{λ x_{2} + λ i y_{2} + x_{3} + i y_{3}}{λ + 1}$

$x_{1} = \frac{λ x_{2} + x_{3}}{λ + 1}$ and $y_{1} = \frac{λ y_{2} + y_{3}}{λ + 1}$

Hence $z_{1}$ (point $A$ ) divides $z_{2}$ (point $B$ ) and $z_{3}$ (point $C$ ) in the ratio $λ : 1$ .
$∴ B, A$ and $C$ are collinear.
$∴$ The distance of point $A$ from line joining $B$ and $C$ is zero.

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If A, B, C are three points in argand plane representing the complex number z1,z2,z3 such that z1=λz2+z3λ+1, where λϵR, then find the distance of point A from the line joining points B and C.

If A, B, C are three points in argand plane representing the complex number $z_{1}, z_{2}, z_{3}$ such that $z_{1} = \frac{λ z_{2} + z_{3}}{λ + 1}$ , where $λ ϵ R$ , then find the distance of point A from the line joining points B and C.