A circle drawn in the rectangular coordinate plane

• has one common point with the line $d_{1}: y - \frac{4x}{3} - 46 = 0$,
• has two common points with the line $d_{2}: y - \frac{4x}{3} - 6 = 0$,
• has no common points with the line $d_{3}: y - \frac{4x}{3} - 1 \geqslant 0$.

It is known that. Accordingly, which of the following could be the radius of this circle in units?
A) 11 B) 13 C) 15 D) 17 E) 19

Let $m$ and $n$ be real numbers. In the rectangular coordinate plane, a circle passing through point $A(4, 1)$ is drawn with equation
$$x^{2} + y^{2} - 2x + 6y = n$$
The line $y = mx$ drawn in the plane intersects this circle at points $B$ and $C$. Given that $m(\widehat{BAC}) = 90^{\circ}$, what is the sum $m + n$?
A) 8 B) 9 C) 10 D) 11 E) 12