Question 112·200 Super-Hard SAT Math Questions·Advanced Math

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For a real number $k$ , consider the system of equations:

\begin{aligned} &x^2+y^2-12x+4y-9=0 \\[1.4em] &4x+3y=k \end{aligned}

Which choice is the greatest possible value of $k$ for which the system has exactly one real solution $(x,y)$ ?

An expanded circle equation paired with a line whose right side can change signals a tangency question: exactly one solution means the line touches the circle once. Graph the circle in Desmos and use its top and bottom points to find its center and radius. Then use the line’s coefficients to find how far its left side can increase. Don’t estimate a barely touching line with the slider.

Hints

Hint 1
For an expanded circle equation, graph it exactly as written. Its highest and lowest points are the ends of a diameter, a segment through the center. What do their midpoint and half their distance tell you?
Hint 2
Each point shared by the circle and line is one solution. Exactly one shared point means the line is tangent: it touches the circle rather than cutting through it. Changing $k$ slides the line without changing its direction.
Hint 3
Start with the value of $4x+3y$ at the circle’s center. A move of one unit in the direction $(4,3)$ increases that expression by the length of $(4,3)$ , which is $5$ . How far can you move from the center?

Step-by-step

Graph the circle, then find the greatest tangent value
Step 1
Read the circle’s vertical endpoints
Type the circle equation as written, without adding $y=$ in front. Select the circle and click its marked highest and lowest points. Desmos shows $(6,5)$ and $(6,-9)$ . They’re the ends of a vertical diameter, a segment across the circle through its center.
Step 2
Find the center and radius
Name those endpoints $A=(6,5)$ and $B=(6,-9)$ . Type $\operatorname{midpoint}(A,B)$ and $\frac{\operatorname{distance}(A,B)}{2}$ . Click the plotted midpoint to see the center $(6,-2)$ ; the distance line prints $7$ . That’s the radius, half the diameter.
Step 3
Connect one solution to a touching line
Type $k=0$ and $4x+3y=k$ . At this starting value, Desmos shows two intersections. Each intersection solves both equations. To get exactly one, the line must be tangent, touching the circle once; a slider can show the change, but it won’t establish the exact value.
Step 4
Get the line’s value at the center
At the center $(6,-2)$ , substitute into the line’s left side. Type $4(6)+3(-2)$ ; Desmos prints $18$ . That’s the center’s value of $4x+3y$ , not the greatest value on the circle.
Step 5
Move one radius in the direction of greatest increase
A point on the circle is a move $(u,v)$ of length $7$ from its center, so its value of $4x+3y$ is $18+4u+3v$ . The direction $(4,3)$ has length $\sqrt{4^2+3^2}=5$ . A length- $7$ move adds at most $7\cdot5$ to the expression, with equality only when the move points in that direction. That one farthest point gives the greatest value and one touching line. Type $18+7\sqrt{4^2+3^2}$ ; Desmos prints $53$ . So the greatest $k$ giving exactly one real solution is $53$ . Choice C.

Question 112·200 Super-Hard SAT Math Questions·Advanced Math

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