Line $l$ in the coordinate plane has 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 $k?$ $\textbf{(A) } 10 \qquad$ $ \textbf{(B) } 15 \qquad$ $ \textbf{(C) } 20 \qquad$ $ \textbf{(D) } 25 \qquad$ $ \textbf{(E) } 30$
Solution
The point $(20,20)$ lies on $l$, since $3\cdot20-5\cdot20+40=0$; so the rotated line $k$ also passes through $(20,20)$.
Let $\alpha$ be the angle between $l$ and the $x$-axis. Then $\tan\alpha=\frac35$ (the slope of $l$).
Let $\beta$ be the angle between $k$ and the $x$-axis. Then $\tan\beta=\tan(45^\circ+\alpha)=\frac{1+\frac35}{1-\frac35}=4$.
Line $l$ has equation $x-3y+24=0$. This line is rotated $45^\circ$ counterclockwise about the point $(12,12)$ to obtain line $k$. What is the $x$-coordinate of the $x$-intercept of line $k$?
