Question

Two lines represented by equations x + y = 1 and x + ky = 0 are mutually orthogonal if k is
A. 1
B. -1
C. 0
D. None of these

Answer 1

Hint:First of all, we will determine the slope of both equations. and we know that If the equations of two lines are perpendicular to each other, then the product of the slope of both the equations will be -1. Hence, we will get a suitable answer.

Formula used:
$y=mx+c$
$m_{1}m_{2}=-1$

Complete step-by-step solution:
Convert both the lines into the general form of the line that is y = mx + c. After that, compare the equations of lines from the general form of the equation of the line. we will get the slope of both the lines. As per the given condition that both the lines are perpendicular to each other. Apply the concept that satisfies the given condition.
Let us assume that Slope of the line $x+y=1$ is $m_{1}$ and $x+ky=0$ is $m_{2}$.
And we know that the general equation of the line is,
$y=mx+c$ ……..(A)
Therefore, Change both the lines into general form.
$x+y=1$
$\Rightarrow~y=-x+1$ ……….(1)
Compare equation (1) from equation (A).
We get,
$m_{1}=-1$ and
$x+ky=0$
$\Rightarrow~y=\dfrac{-x}{k}+0$ …….(2)
Compare equation (2) to equation (A).
we get,
$m_{2}=\dfrac{-1}{k}$
Lines are mutually perpendicular. So the product of the slope of the lines will be equal to -1.
Therefore,
$m_{1}m_{2}=-1$
$\Rightarrow~-1\lgroup\dfrac{-1}{k}\rgroup=-1$
$\Rightarrow~k=-1$

Option B. is correct.

Note: It is important to compare the given line with the general form of the line $y=mx+c$. Before comparing the equation of the lines with y = mx + c, convert both the equations of the lines into the general form of the equation of a line.