Suppose that ω and z are complex numbers such that both (1+2 i) ω and (1+2 i) z are different real numbers. The slope of the line connecting ω and z in the complex plane is

Question

Suppose that ω and z are complex numbers such that both (1+2 i) ω and (1+2 i) z are different real numbers. The slope of the line connecting ω and z in the complex plane is (A) -2 (B) -1 / 2 (C) 2 (D) can not be determined

Tardigrade Admin · Accepted Answer

Let ω= x 1+i y 1 and z = x 2+i y 2 now (1+2 i)(x1+i y1) is real ⇒ 2 x 1+ y 1=0 Illy 2 x 2+ y 2=0 ∴ 2 x 1+ y 1=2 x 2+ y 2 -2( x 2- x 1)= y 2- y 1 ( y 2- y 1/ x 2- x 1)=-2

Q. Suppose that $ω$ and $z$ are complex numbers such that both $(1 + 2 i) ω$ and $(1 + 2 i) z$ are different real numbers. The slope of the line connecting $ω$ and $z$ in the complex plane is

-2

-1 / 2

2

can not be determined

Let $ω = x_{1} + i y_{1}$ and $z = x_{2} + i y_{2}$
now
$(1 + 2 i) (x_{1} + i y_{1})$ is real
$\Rightarrow 2 x_{1} + y_{1} = 0$
$I ll y 2 x_{2} + y_{2} = 0$
$∴ 2 x_{1} + y_{1} = 2 x_{2} + y_{2}$
$- 2 (x_{2} - x_{1}) = y_{2} - y_{1}$
$\frac{y _{2} - y _{1}}{x _{2} - x _{1}} = - 2$

Q. Suppose that ω and z are complex numbers such that both (1+2i)ω and (1+2i)z are different real numbers. The slope of the line connecting ω and z in the complex plane is

-2

-1 / 2

2

can not be determined

Let ω=x1​+iy1​ and z=x2​+iy2​ now (1+2i)(x1​+iy1​) is real ⇒2x1​+y1​=0 Illy2x2​+y2​=0 ∴2x1​+y1​=2x2​+y2​ −2(x2​−x1​)=y2​−y1​ x2​−x1​y2​−y1​​=−2

Q. Suppose that $ω$ and $z$ are complex numbers such that both $(1 + 2 i) ω$ and $(1 + 2 i) z$ are different real numbers. The slope of the line connecting $ω$ and $z$ in the complex plane is

Let $ω = x_{1} + i y_{1}$ and $z = x_{2} + i y_{2}$
now
$(1 + 2 i) (x_{1} + i y_{1})$ is real
$\Rightarrow 2 x_{1} + y_{1} = 0$
$I ll y 2 x_{2} + y_{2} = 0$
$∴ 2 x_{1} + y_{1} = 2 x_{2} + y_{2}$
$- 2 (x_{2} - x_{1}) = y_{2} - y_{1}$
$\frac{y _{2} - y _{1}}{x _{2} - x _{1}} = - 2$