Question 660·Hard·Math

The line must be tangent, meaning it touches the circle at exactly one point. Count shared points, not how close a line looks to the circle. A line that cuts through has two; one that misses has none.

Type the circle equation as written. Then type $y=2x+1$ to test $m=2$, keeping the line's given $y$-intercept of $1$. Desmos shows no shared point, so this line misses the circle.

Add $y=\frac{\sqrt{65}-9}{4}x+1$. Desmos shows two intersections with the circle. That line crosses the circle rather than touching it once.

Add $y=\frac{9+\sqrt{65}}{8}x+1$. Desmos shows no intersection for this line either. A line can pass near a circle without touching it.

Add $y=\frac{9-\sqrt{65}}{4}x+1$ and click where it meets the circle. Desmos shows about $(2.177,1.510)$, with one shared point. So $\frac{9-\sqrt{65}}{4}$ could be the slope $m$. Choice D.

At a shared point, $y=mx+1$ must satisfy the circle equation. Replace $y$ with $mx+1$:

Combine $1+2$ inside the second square:

Expand both squares, treating $mx+3$ as the whole second quantity:

Collect like terms:

Subtract $13$:

Each solution for $x$ gives a point where the line meets the circle. For a quadratic $ax^2+bx+c=0$, the discriminant is $b^2-4ac$. It must be zero for exactly one real solution, because the plus and minus parts of the quadratic formula then give the same value:

Expand the square and product:

Combine like terms:

Divide every term by $8$:

To solve the slope equation in Desmos, call the slope $x_1$: Desmos treats $x_1$ as a value to find. Type $2x_1^2-9x_1+2\sim0\{x_1<1\}$. The $\sim$ asks Desmos to fit the equality, and the restriction selects the root below $1$. Under PARAMETERS, Desmos reports $x_1\approx0.2344$.

Use the quadratic formula on $2m^2-9m+2=0$ to keep the roots exact:

Type $\frac{9-\sqrt{65}}{4}$ below the regression. Desmos prints about $0.2344$, matching its smaller root. The other root isn't among the choices, so $m=\frac{9-\sqrt{65}}{4}$. Choice D.