## Find equation of a parabola when neither of two points is the vertex

Can we find the equation of a parabola when two points on it and the time of travel between the two points are given.It is also given that neither of the two given points is the vertex.
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Recognitions:
 Recognitions: Gold Member Science Advisor Staff Emeritus Assuming that the parabola has vertical axis, then the equation can be written $y= ax^2+ bx+ c$. You need three conditions to solve for the three coefficients, a, b, and c. Saying that the given point $(x_0,y_0)$ is on the parabola means that $y_0= ax_0^2+ bx_0+ c$. Saying that the given point $(x_1,y_1)$ is on the parabola means that $y_1= ax_1^2+ bx_1+ c$. That gives you two of the three equations you need. If $y= ax^2+ bx+ c$, then $y'= 2ax+ b$ so the arc-length between $x_0$ and $x_1$ is given by $$\int_{x_0}^{x_1}\sqrt{1- (2ax+ b)^2}dx$$ If you know the arc-length between the two points, you would have the third. Calculate the formula for arclength between the two points and set equal to that. Of course, if you know the "uniform velocity", the arclength is just the time divided by that velocity. If you don't know the velocity, then, as mathman said, you don't have enough information to determine the parabola.