20 Let z1 and z2 be distinct complex numbers Show that the l

20. Let z1 and z2 be distinct complex numbers. Show that the locus of points tz1 + (1 - t)z2, -infinity

Solution

Let z₁ = x₁ +iy₁ and z₂ = x₂+iy₂

tz₁ + (1-t)z₂ = t(x₁+iy₁) + (1-t)(x₂+iy₂) = (tx₁+(1-t)x₂)+i(ty₁+(1-t)y₂)

Clearly the points are of the form (a,b) , where a = tx₁+(1-t)x₂ and b = ty₁+(1-t)y₂

So a and b are real numbers

And this is a line, Now if t=0, then tz₁ + (1-t)z₂ = z₂

And if t=1, then  tz₁ + (1-t)z₂ = z₁

And so this line passes through z₁ and z₂