A squared variable is a cue that a system may have more than one solution. Graph both equations in Desmos and click every intersection, since each one satisfies both equations. If the question asks what a value must be, check it at every point. Using only one intersection might miss a different result at another.
Hints
- Hint 1
In a system, a solution must satisfy both equations. Graph each equation as written, then click every point where the line meets the circle. How many solutions do you find?
- Hint 2
A product is the result of multiplication. At each intersection, multiply the first coordinate, , by the second, . Why isn't checking only one point enough to show what must be?
Step-by-step
Approach 1: Graph both equations and check each product
Step 1Find every solution
Type and on separate lines. The first graph is a circle and the second is a line. Click both intersections: Desmos shows and . Each point lies on both graphs, so each solves the system.
- Step 2
Check the product at both points
Type and to multiply the coordinates of each solution. Desmos prints for both; the two negatives in the first product make it positive. To show a value must hold, check every solution, not just one. So for either solution. Choice C.
Approach 2: Find the product without finding the points
Step 1Get the difference between the coordinates
The line says . Subtract from both sides to get the difference between the coordinates:
- Step 2
Square the difference
Square both sides to get squared terms like the ones in the circle equation:
- Step 3
Expand the squared difference
Expand using the square of a difference, :
The term appears because the expansion multiplies and .
- Step 4
Use the circle equation
The circle gives , so replace the two squared terms in the expanded equation with :
- Step 5
Solve for the required product
Subtract from both sides:
Divide both sides by :
The equations fix the same product for every solution, even though the coordinates differ. Choice C.