# Solve equation for x in d1+d2 =d1*d2

Im trying to solve one equation which I will be using in one software.

I have two fixed points a and b in 2D space and also point c in which i know just coordinate x .

I need to solve coordinate y for point c. This equation can have more solutions. In this equation must be true that

s2*d1 + s1+d2 =d1*d2

where: d1 = squared distance between points a and c d1 = squared distance between points b and c s1 = variable which must be contained in equation s2 = variable which must be contained in equation

Here is the link for Desmos: https://www.desmos.com/calculator/g85twlnqcx

If I try to understand, you have two numbers $$(s_1,s_2)$$, the coordinates $$(x_a,y_a)$$ for point $$A$$, $$(x_b,y_b)$$ for point $$B$$, $$(x_c,y_c)$$ for point $$C$$ and you want to solve for $$y_c$$ the equation $$s_2 d_1+s_1 d_2=d_1 d_2$$ where $$d_1=(x_a-x_c)^2+(y_a-y_c)^2 \qquad \text{and} \qquad d_2=(x_a-x_c)^2+(y_a-y_c)^2$$ Expanding, you will end with a quartic equation in $$y_c$$ $$y_c^4+\alpha \,y_c^3+\beta\, y_c^2+\gamma \,y_c+\delta=0$$ where $$\alpha=-2(y_a+y_b)$$ $$\beta=-2 x_c (x_a+x_b)+4 y_a y_b+x_a^2+y_a^2+x_b^2+y_b^2+2 x_c^2-s_1-s_2$$ $$\gamma=-2 y_a \left(-2 x_b x_c+x_b^2+y_b^2+x_c^2\right)-2 y_b \left(-2 x_a x_c+x_a^2+y_a^2+x_c^2\right)+2 s_2 y_a+2 s_1 y_b$$ $$\delta=\left(\left((x_a-x_c)^2+y_a^2\right) \left((x_b-x_c)^2+y_b^2\right)+2 s_2 x_a x_c-s_2 x_a^2-s_2 y_a^2+2 s_1 x_b x_c-s_1 x_b^2-s_1 y_b^2-s_1 x_c^2-s_2 x_c^2\right)$$