Show that the following system of parametric equations describes a line or a parabola:
$$\begin{cases} x=a_1t^2+b_1t+c_1 \\ y=a_2t^2+b_2t+c_2 \end{cases}, t\in\mathbb{R}.$$
[Math] Parabola in parametric form
conic sectionsgeometryparametric
conic sectionsgeometryparametric
Show that the following system of parametric equations describes a line or a parabola:
$$\begin{cases} x=a_1t^2+b_1t+c_1 \\ y=a_2t^2+b_2t+c_2 \end{cases}, t\in\mathbb{R}.$$
Best Answer
You can eliminate the parameter $t$: Subtraction of the two equations gives: $$a_2 x-a_1 y=(a_2b_1-a_1b_2)t+(a_2c_1-a_1c_2)$$ Multiply first equation by $(a_2b_1-a_1b_2)^2$: $$(a_2b_1-a_1b_2)^2x=a_1 ((a_2b_1-a_1b_2)t)^2 +b_1 (a_2b_1-a_1b_2)^2 t+(a_2b_1-a_1b_2)^2c_1$$ from which: $$(a_2b_1-a_1b_2)^2x=a_1 (a_2 x-a_1 y-(a_2c_1-a_1c_2))^2 +b_1 (a_2b_1-a_1b_2)^2(a_2 x-a_1 y-(a_2c_1-a_1c_2))+(a_2b_1-a_1b_2)^2 c_1.$$ Rewrite in the form: $$ax^2+bxy+cy^2+ux+vy+w=0$$ then it's enoght to check $b^2-4ac=0$ to verify that this is a parabola (provided that $a,b,c$ not all $0$).