Question

A line passes through the origin and makes an angle of $\pi /4$ with the line $x-y+1=0$. Then.

• A. Equation of the line is $x=0$
• B. The equation of the line is $y=0$
• C. The point of intersection of the line with the given line is $(-1,0)$
• D. The point of intersection of the line with the given line is $(0,1)$

Answer 1

Answer: (B), (C)

The point of intersection of the line with the given line is $(-1,0)$
The equation of the line is $y=0$

Let the line which passes through origin and make an angle $\pi /4$ with the line $x-y+1=0$

is $AB$

Let slope of line $AB=m_2$

Let slope of line $CD(x-y+1=0)=m_1=1$

It is given that the angle between $AB$ and $CD$ is ${45}^o$

$\tan \theta =\left|\frac{m_1+m_2}{1+m_1{m}_2}\right|$

$\tan {45}^o=\left|\frac{1-m_2}{1+m_2}\right|$

$\pm 1=\left(\frac{1-m_2}{1+m_2}\right)$

$1=\frac{1-m_2}{1+m_2},-1=\left(\frac{1-m_2}{1+m_2}\right)$

$1+m_2=1-m_2$ $-(1+m_2)=(1-m_2)$

$2m_2=0$ $-1-m_2=1--m_2$

$m_2=0$ $-2=0$ (not possible)

$\therefore m_2=0$

$\therefore$ Equation of line $(y-y_1)=m(x-x_1)$

$\Rightarrow (y-0)=0(x-0)$

$y=0$

To find: Point of intersection of the line with the given line

Put $y=0$ in the given line equation

$x-0+1=0$

$x=-1$

$\therefore$ Point of intersection is $(-1,0)$