Question 102·Hard·Nonlinear Equations in One Variable; Systems in Two Variables

0 / 233

Consider the system of equations

\begin{cases}(x-k)^2 + y^2 = 4\\[4pt]y = 2x + 1\end{cases}

where $k$ is an integer constant.
If the system has no real solution, what is the least positive integer value of $k$ ?

Your answer

In a system with a circle and a line, no intersection means no real solution. Graph both equations in Desmos and change the integer constant to see when the graphs stop meeting. Start with the smallest positive integer and work upward. Before deciding the graphs miss, make sure the whole circle is visible; an intersection could otherwise be offscreen.

Hints

Hint 1
A solution to a system is a point that satisfies both equations, so it appears where their graphs meet. No real solution means the line and circle have no shared point.
Hint 2
The least positive integer is $1$ . Test $k=1$ first; if the graphs meet, change $k$ to the next positive integer.

Step-by-step

Test positive integers on the graph
Step 1
Identify the circle you need to see
Rewrite the first equation as $(x-k)^2+y^2=2^2$ . This is a circle centered at $(k,0)$ with radius $2$ . Changing $k$ moves the circle, but the line stays fixed. Keep the whole circle in view when checking for intersections.
Step 2
Test the smallest positive integer
Start with $k=1$ . Type $k=1$ , $(x-k)^2+y^2=4$ , and $y=2x+1$ on separate Desmos lines. The graphs cross twice. Each intersection satisfies both equations, so $1$ does not give no real solution.
Step 3
Test the next integer and choose the least
Change the first Desmos line to $k=2$ . The line now misses the entire circle, which is visible in the graph window. So the system has no real solution. Since $1$ failed and there is no positive integer between $1$ and $2$ , the least value of $k$ is $2$ . Grid in 2.

Question 102·Hard·Nonlinear Equations in One Variable; Systems in Two Variables

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this