We have a circumference that is given by the following equation:
[tex]x^{2}+y^{2}=49[/tex]
We can write this equation in its standard form as follows:
[tex]x^{2}+y^{2}=7^{2} \\ where \ the \ radius \ r=7[/tex]
On the other hand, the linear function is given as the following table:
[tex]x \ \ \ \ \ \ \ \ g(x) \\ -1 \ \ -9.2 \\ 0 \ \ \ \ \ -9 \\ 1 \ \ \ \ \ -8.8[/tex]
To check if the circle and the line intersects, let's substitute the equation of the line into the equation of the circle to see if there is a real solution, so:
[tex]x^{2}+(0.2x-9)^{2}=49 \\ \\ \therefore x^{2}+0.04x^{2}-3.6x+81=49 \\ \\ \therefore 1.04x^{2}-36x+32=0 \\ \\ Solving \ for \ x: \\ x_{1}=33.70 \\ x_{2}=0.91 \\ \\ Solving \ for \ y: \\ y_{1}=0.2(33.70)-9=-2.26 \\ y_{2}=0.2(0.91)-9=-8.18[/tex]
Finally the intersects are:
[tex]P_{1}(33.70, -2.26) \ and \ P_{2}(0.91, -8.18)[/tex]
