Convention in parametric equation of parabola.

In summary, the general equation of a parabola is y^2 = 4ax, and its parametric equation is ((at^2), (2at)). However, other parametric equations such as ((t^2)/4a, t) can also define a parabola easily. It seems that using ((at^2), (2at)) as the parametric equation for a parabola is a convention rather than a specific reason. Some questions may be based on this equation, so it is important to confirm if it is a convention or if there is another reason for its use.
  • #1
vkash
318
1
general equation of parabola is y^2=4*a*x. it's parametric equation is ((a*t^2),(2*a*t)) [as in my book] but i think there can be other kind of parametric equations also like( ((t^2)/4*a),t) it defines a parabola easily. is using ((a*t^2),(2*a*t)) as parametric equation of parabola is convention or there is any reason for this.
I think it's a convention but some question are specially based on this equation ((a*t^2),(2*a*t)) so i want to confirm is it really a convention or there is any other thing.

thanks for reading. write your view.
 
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  • #2
Hi vkash! :smile:

There aren't really any good reasons for taking (at2,2at) as parametrization. The only good reason that I can think of is that we don't work with fractions with this parametrization, and fractions are sometimes annoying.
 
  • #3
micromass said:
Hi vkash! :smile:

There aren't really any good reasons for taking (at2,2at) as parametrization. The only good reason that I can think of is that we don't work with fractions with this parametrization, and fractions are sometimes annoying.
thanks genius.
 

FAQ: Convention in parametric equation of parabola.

1. What is the convention for parametric equations of parabolas?

The convention for parametric equations of parabolas is to use the parameter t and the standard form of the equation, x = at^2 + bt + c, y = dt^2 + et + f, where a, b, d, e, and f are constants.

2. How do you determine the direction of a parabola using parametric equations?

The direction of a parabola can be determined by the sign of the coefficient a in the x equation. If a is positive, the parabola opens to the right. If a is negative, the parabola opens to the left.

3. Can parametric equations of parabolas be used to find the vertex?

Yes, the vertex of a parabola can be found by substituting the value of t = -b/2a into the x and y equations. This will give the coordinates of the vertex as (x, y).

4. What is the significance of the parameter t in parametric equations of parabolas?

The parameter t represents the independent variable in the equations and is used to determine the points along the parabola. It can take on any real value and is often used to create a graph of the parabola.

5. Are there any other conventions for parametric equations of parabolas?

Some other conventions for parametric equations of parabolas include using different variables for the parameters (such as x = aθ^2 + bθ + c, y = dθ^2 + eθ + f), using trigonometric functions in the equations (such as x = acosθ + b, y = asinθ + c), or using a combination of both methods.

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