Question

How many points do you need to determine the equation of a parabola?

Answer 1

Hint: To solve this question, we need to consider the general equation of a parabola. The number of independent constants which are present in the standard equation are to be determined for the determination of the equation of a parabola. The minimum points lying on the parabola needed for determining the values of these independent constants will be equal to the final answer.

Complete step-by-step solution:
Let us consider that the axis of the parabola is vertical. We know that the standard equation of such a parabola is given by
$\left( {y - k} \right) = a{\left( {x - h} \right)^2}$
Here $\left( {h,k} \right)$ are the coordinates of the vertex of the parabola, and $a$ is a constant. So overall we have three independent constants appearing in the standard equation of a parabola. For determining the equation of a parabola, we have to solve for these three independent constants. So we need minimum three equations in terms of a, h, and k.
For obtaining three equations in terms of the three independent constants a, h and k, we need three points lying on the parabola.

Hence, for determining the equation of a parabola, we need a minimum number of three points.

Note:
We may argue that for the case of the parabola represented by the standard equation ${y^2} = 4ax$, we have only one independent constant appearing in this equation and therefore only one point is needed to determine its equation. But we must note that ${y^2} = 4ax$ is not the general equation of a parabola. It is the special case of a parabola whose vertex lies at the origin.