Question

The equations of the lines through $\left(1,1\right)$ and making angles of $45°$ with the line $x+y=0$ are

• A. $x-1=0,x-y=0$
• B. $x-y=0,y-1=0$
• C. $x+y-2=0,y-1=0$
• D. $x-1=0,y-1=0$

Answer 1

The correct option is D

$x-1=0,y-1=0$

Step 1: Calculate the slope

The given equation of line is $x+y=0$.

$y=-x$

Slope of the line, $m_1=-1$.

Let $m_2=m$.

Angle between two lines is calculated as $\tan \theta =\frac{m_2-m_1}{1+m_1{m}_2}$.

$\begin{array}{rcl}\tan 45°& =& \frac{\left(m+1\right)}{1-m}\\ 1& =& \frac{\left(m+1\right)}{1-m}\\ 1-m& =& m+1\\ 2m& =& 0\\ m& =& 0\end{array}$

and

$\begin{array}{rcl}1& =& \frac{m+1}{m-1}\\ & \Rightarrow & m=\infty \end{array}$

Step 2: Calculate the equation of line

Equation of line passing through $\left(1,1\right)$ is $\left(y-y_1\right)=m\left(x-x_1\right)$

When $m=0,\left(y-1\right)=0$

When $m=\infty ,\left(x-1\right)=0$

Hence, the equations of the lines are $x-1=0,y-1=0$ and therefore option (D) is correct.