A circle and a line with a changing constant form a tangency question when the system must have exactly one shared point. Graph the system in Desmos to see how intersections count as solutions. Then use the fact that a radius is perpendicular to a line where it touches the circle. A line that cuts through the circle has two solutions, not one.
Hints
- Hint 1
A solution is a point on both graphs. Exactly one solution means the line touches the circle at one point rather than crossing it twice. What happens when you graph the line with one value of ?
- Hint 2
A radius runs from the circle’s center to a point on it. At a touch point, that radius is perpendicular to the line. The line’s slope is , so what slope must the radius have?
- Hint 3
The circle is centered at , so a radius with slope lies on . Graph that line to find the possible touch points, then put each point into .
Step-by-step
Find where the line can touch the circle
Step 1Connect intersections to solutions
Type , , and in Desmos. The line crosses the circle twice. Each shared point solves both equations, so this setting gives two solutions, not one.
- Step 2
Locate the possible touch points
The circle’s center is because is the squared distance from the origin. At a tangent point, where a line touches a circle, the radius is perpendicular to the line. The line has slope , so that radius has slope and lies on . Add in Desmos and click its two intersections with the circle. Desmos shows and , the possible touch points.
- Step 3
Find the constant for each touch point
Each touch point must also satisfy . Subtract to get the constant:
Use :
The other point gives , which isn’t offered. So the listed value that makes the line touch the circle exactly once is . Choice B.