Question

The line $4x-3y+2=0$ is rotated through an angle of $\frac{\pi }{4}$ in clockwise direction about the point $(1,2).$ The equation of the line in its new position is

• A. $x-7y+13=0$
• B. $y-7x+5=0$
• C. $x+7y-15=0$
• D. $y+7x-15=0$

Answer 1

The correct option is A $x-7y+13=0$


$(1,2)$ lies on the line $4x-3y+2=0$
The slope of the given line is $\tan \theta =\frac{4}{3}$
Let the slope of the new line be $m.$
Angle between these two lines is ${45}^{\circ }$
$\Rightarrow \tan {45}^{\circ }=\left|\frac{\frac{4}{3}-m}{1+\frac{4}{3}m}\right|$
$\Rightarrow \frac{4-3m}{3+4m}=\pm 1$
$\Rightarrow 4-3m=\pm (3+4m)$
$\Rightarrow m=\frac{1}{7}(m=-7\text{rejected})$

Required equation of line is
$y-2=\frac{1}{7}(x-1)$
$\Rightarrow x-7y+13=0$