Question

50 points
why do we take any point on a parabola as (at^2 ,2at) ?here t is a parameter.

Answer 1

The best and easiest form to represent the co-ordinates of any point on the parabola y2 = 4ax is (at2, 2at).

Since, for all the values of ‘t’ the coordinates (at2, 2at) satisfy the equation of the parabola y2 = 4ax.

Standard equation of the parabola y2 = -4ax:

Parametric coordinates of the parabola y2 = -4ax are (-at2, 2at).

Parametric equations of the parabola y2 = -4ax are x = -at2, y = 2at.

Standard equation of the parabola x2 = 4ay:

Parametric coordinates of the parabola x2 = 4ay are (2at, at2).

Parametric equations of the parabola x2 = 4ay are x = 2at, y = at2.

Standard equation of the parabola x2 = -4ay:

Parametric coordinates of the parabola x2 = -4ay are (2at, -at2).

Parametric equations of the parabola x2 = -4ay are x = 2at, y = -at2.

Standard equation of the parabola (y - k)2 = 4a(x - h):

The parametric equations of the parabola (y - k)2 = 4a(x - h) are x = h + at2 and y = k + 2at.

Hope it helps

Answer 2

Answer:

Explanation:

Answer: we take any point on a parabola as (at^2 ,2at) easiest form to represent the co-ordinates of any point on the parabola y2 = 4ax is (at2, 2at). Since, for all the values of ‘t’ the coordinates (at2, 2at) satisfy the equation of the parabola y2 =  4ax.