Question

"Find the point at which the line touches the circle" I really need help no one helped last time...

Answer 1

The solution for a system of equations will be when the graphs touch each other, or intersect, and that happens when both equations equate each other, so, let's do so,

$\bf \begin{cases} \boxed{y}=3x+10\\ -------\\ x^2+y^2=10\\ y^2=10-x^2\\ y=\sqrt{10-x^2} \end{cases}\qquad \qquad \boxed{3x+10}=\sqrt{10-x^2} \\\\\\ \textit{now we square both sides}\qquad (3x+10)^2=(\sqrt{10-x^2})^2 \\\\\\ (3x+10)^2=10-x^2\implies 9x^2+60x+100=10-x^2 \\\\\\ 10x^2+60x-90=0\implies x^2+6x-9=0\implies (x+3)(x+3)=0 \\\\\\ x= \begin{cases} -3\\ -3 \end{cases}\qquad thus\qquad \boxed{x=-3}$

$\bf -------------------------------\\\\ \textit{now, when x = -3, what is \underline{y}?}\qquad y=3x+10\implies y=3(-3)+10 \\\\\\ y=-9+10\implies \boxed{y=1}\qquad \qquad therefore\qquad (\stackrel{x}{-3}~,~\stackrel{y}{1})$