A circle and a parabola given by equations form a nonlinear system: a shared point makes both equations true. When the question asks how many points they share, graph both equations in Desmos and count their intersections. Keep the whole circle in view, and don't mistake its center or other marked points for intersections.
Hints
- Hint 1
In center-radius form, , the center is and the radius is . The signs inside the parentheses reverse when you read the center. Where is this circle centered?
- Hint 2
A circle's equation draws only its edge, not everything inside it. Graph the circle exactly as written in Desmos, without putting in front. Make sure its full edge fits in the window.
- Hint 3
An intersection lies on both graphs. Add the parabola on its own Desmos line and click each place the curves meet. Don't count a marked point that belongs to only one graph.
Step-by-step
Graph both equations and count shared points
Step 1Find the circle's center and radius
In center-radius form, , the center is and is the radius. Here and , so the center is and the radius is .
- Step 2
Graph the entire circle
Type on one Desmos line as written, without adding . Desmos draws the circle. Select it and click its marked top and bottom points to see and . The window shows its whole edge, so a meeting point can't hide offscreen.
- Step 3
Add the parabola and count its crossings
Type on the next line and click the places where it meets the circle. Desmos shows crossings near and . The parabola's vertex, its lowest point, is , but that's the circle's center, not a point on its edge. Count only points shared by both graphs. They intersect at exactly two points. Choice B.
Lessons that teach this
- SAT Advanced AlgebraAdvancedCoreSolve linear and nonlinear systems
- SAT Advanced AlgebraIntermediateUse the quadratic formula and discriminant
- SAT Geometry and TrigonometryBeginnerRead and write circle equations
- DesmosIntermediateCoreHow many solutions?
- DesmosIntermediateCoreCircle equations in the graph