Circle Quadratic System of Equation problem

Question

x²+y²=25

y=x²-13

If possible, rewrite the system shown above, keeping one of the equations the same but changing the other equation to satisfy each of the following stated requirements. If the change is not possible, explain why.

a) There are exactly three solutions for the system

b) There are exactly two solutions for the system

c) There is only one solution for the system

d) There is no only solution for the system for the system

Answer 1

Start by plotting the functions using a graphing calculator or an online tool such as DESMOS.

As written, there are 4 solutions to the pair of equations. Each of the 4 conditions can be met by doing a vertical translation of the second equation.

3 solutions: put the origin of the second equation at y = -5

2 solutions: put the origin of the second equation at any value > - 5, but < +5

1 solution: put the origin of the second equation at x = +5

NO solutions: put the origin of the second equation at x > +5

You can also get 2 or 0 solutions by moving the second curve DOWN