Question

The line $x+2y=4$ is translated parallel to itself by $3$ units in the sense of increasing $x$ and is then rotated by ${30}^o$ in clockwise direction about the point where the shifted line cuts the $x$-axis. Find the equation of line in new position.

Answer 1

Given a line x+2y=4 is translated paralleled by 3 units in 30° slope of the line is same as that of $x+2y=4$

$\therefore$ Slope of new line =$1$ and $\theta =30°$

Let $A(x,y)$ be the initial point and $A^{\prime }(x^{\prime },y^{\prime })$ be the final point. The line along which the point has been moved

$y=\frac{4-x}{2}$

It has been moved to a distance of 3 units.The values of the equation are $(0,2),(4,0)$ etc.

By using simple trigonometry ,the distance along the x-axis which the line has moved is $r\cos \theta$ (where $\theta$ is the angle the line makes with x-axis)

$\therefore$ The distance moved by y-axis is $r\sin \theta$

$\therefore$ we have 2 new equations

$x^{\prime }+r\cos \theta =x{y}^{\prime }+r\sin \theta =y7+3\cos 30=x,5+3\sin 30=y\therefore x=26.92,y=6.5$

$\therefore$ The new equation

$y-y_1=m(x-x_1)\Rightarrow y-6.5=1(x-26.92)\Rightarrow x-y+6.5-26.92=0\therefore x-y-20.42=0$