2020
Problem - 4737
Line $\ell$ in the coordinate plane has the equation $3x - 5y + 40 = 0$. This line is rotated $45^{\circ}$ counterclockwise about the point $(20, 20)$ to obtain line $k$. What is the $x$-coordinate of the $x$-intercept of line $k$?
First, we note that $(20, 20)$ is on this line $\ell$. Hence, all it take is to determine the slope of like $k$.
Let the angle of the line $\ell$ against the $x$-axis be $\alpha$, then $\tan\alpha=\frac{3}{5}$ which is the slope of $\ell$. If follows that the slope of the line $k$ equals: $$\tan(\alpha+45^{\circ})=\frac{\tan\alpha + \tan 45^{\circ}}{1-\tan\alpha\tan 45^{\circ}}=4$$
From here, it is easy to find the equation of the line $k$ is $y=4x - 60$ whose $x$-intercept is $\boxed{15}$.