Question 1006·Medium·Math

No Desmos needed. The circle’s top gives you the tangent direction. In $x^2+y^2=r^2$, the center is $(0,0)$ and $r$ is the radius, the distance from the center to the circle. Since $25=5^2$, the radius is $5$, so the highest point is $(0,5)$.

At $x=0$, the line gives $y=k(0)+5=5$ for every $k$. So every line in the question already meets the circle at $(0,5)$. The trap is stopping there: you still need to know whether it meets the circle again.

A tangent touches a circle at exactly one point and is perpendicular to the radius there. The radius from $(0,0)$ to $(0,5)$ is vertical, so the tangent at the top must be horizontal. A tilted line through the top enters the circle and comes out at a second point. Since $k$ is the line’s slope, or rise per unit to the right, horizontal means $k=0$. That gives exactly one intersection. Choice C.