Translate line vertically and calculate intersection on circle

Question

Let's say I have a line extending from the center of the circle at a 45° angle. If I were to translate that line up 212.132 units, how would I calculate the intersection between the translated line and the green circle?

Answer 1

Assuming the centre of your circle is the origin, then the equation for this circle is $$x^2+y^2\approx565^2\approx319225$$ $$y^2=319225-x^2$$ You have a $45$ degree angle, meaning a slope of $-1$, forming a line of $$y=-x+212.132$$ Squaring both sides, you get $$y^2=(-x+212.132)^2$$ Now sub in your circle to equate them. $$319225-x^2=(-x+212.132)^2$$ $$319225-x^2=x^2-424.264x+45000$$ $$0=2x^2-424.264x-274225$$ Plug into the quadratic formula: $$x=\frac{424.264\pm\sqrt{180000-4(2)(-274225)}}{2(2)}$$ $$x=\frac{424.264\pm\sqrt{2373800}}{4}$$ $$x=\frac{424.264\pm1540}{4}$$ We're looking for the left side, or negative $x$, so we need the negative. $$x=\frac{424-1540}{4}=-278.934$$ Plugging that into the line earlier, $$y=278.934+212.132=491.066$$ $$(x,y)=(-278.934,491.066)$$ with the centre of the circle as the origin. Note that I rounded a lot of these, but you can plug your numbers into the above to get an exact answer.