Question 185·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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\begin{aligned} x^2 + y^2 &= r^2 \\[1.4em] y &= r \end{aligned}

In the system of equations above, $r$ is a positive constant. How many points of intersection do the graphs of the two equations have in the $xy$ -plane?

A horizontal line can cross a circle, touch it, or miss it. Read the radius, the distance from the center to the circle's edge. Compare the line's height with the center's $y$ -coordinate plus or minus the radius. At the top or bottom, the line is tangent, touching rather than cutting through. Don't assume every horizontal line crosses twice.

Hints

Hint 1
The radius is the distance from a circle's center to its edge. In $x^2+y^2=r^2$ , the center is $(0,0)$ and the radius is positive. What is the circle's highest $y$ -coordinate?
Hint 2
A tangent is a line that touches a circle without cutting through it. The equation $y=r$ keeps every point at the same height. How does that height compare with the circle's highest point?

Step-by-step

Find where the line touches the circle
Step 1
Locate the top of the circle
No Desmos needed. The circle's center and radius give its highest point directly.

In standard form, $x^2+y^2=r^2$ describes a circle centered at $(0,0)$ with radius $r$ . Because $r>0$ , the circle reaches $r$ units above its center, so its highest point is $(0,r)$ .
Step 2
Count the shared points
The line $y=r$ runs through $(0,r)$ , the circle's top. Any other point on that line would be left or right of $(0,r)$ and farther than $r$ from the center, so it couldn't be on the circle. The line is tangent: it touches at exactly one point. The graphs have one point of intersection. Choice B.

Question 185·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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