How do you determine the intersection between parametric curves $(t^{2} + 2 t, - 3 t^{2} + 5 t)$ $(t^2+2t,-3t^2+5t)$ and $(- 2 t^{2} + 4 t, t^{2} + 2 t)$ $(-2t^2+4t,t^2+2t)$ ?

Question

How do you determine the intersection between parametric curves $(t^{2} + 2 t, - 3 t^{2} + 5 t)$ $(t^2+2t,-3t^2+5t)$ and $(- 2 t^{2} + 4 t, t^{2} + 2 t)$ $(-2t^2+4t,t^2+2t)$ ?

As mentioned in the title, I have the following parametric curves:

Red: $(t^{2} + 2 t, - 3 t^{2} + 5 t)$ $(t^2+2t,-3t^2+5t)$
Blue: $(- 2 t^{2} + 4 t, t^{2} + 2 t)$ $(-2t^2+4t,t^2+2t)$

I have tried to write the curves in function form using the method that PatrickJMT demonstrates, but I encountered two $\pm$ $+-$ symbols in the same equation, and I am unsure how to proceed.

How do I find the exact intersections?

Thanks!

1 Answer

Impact of this question

5840 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-10-08T15:58:56

See below.

Explanation:

Defining

$f_{1} (x (t), y (t)) = {t^{2} + 2 t, - 3 t^{2} + 5 t}$ $f_1(x(t),y(t))={t^2 + 2 t, -3 t^2 + 5 t}$ and
$f_{2} (x (t), y (t)) = {- 2 t^{2} + 4 t, t^{2} + 2 t}$ $f_2(x(t),y(t))={-2 t^2 + 4 t, t^2 + 2 t}$

We can verify that the set of coincidence points only has one element: The point $(0, 0)$ $(0,0)$ at the instant $t = 0$ $t = 0$ .

The functions

$f_{1} (x, y)$ $f_1(x,y)$ and $f_{2} (x, y)$ $f_2(x,y)$ have crosses in two points:

${0, 0}$ ${0,0}$ and ${1.8636, 2.02359}$ ${ 1.8636, 2.02359}$

Those points are the solutions of the nonparametric curves

$f_{1} (x, y) = \sqrt{1 + x} - \frac{1}{6} (5 - \sqrt{25 - 12 y}) - 1 = 0$ $f_1(x,y)=sqrt[1 + x] - 1/6 (5 - sqrt[25 - 12 y])-1=0$ and
$f_{2} (x, y) = \frac{1}{2} (2 - \sqrt{2} \sqrt{2 - x}) - (\sqrt{1 + y} - 1) = 0$ $f_2(x,y)=1/2 (2 - sqrt[2] sqrt[2 - x]) - ( sqrt[1 + y]-1)=0$

Science

Math

Humanities

... and beyond

How do you determine the intersection between parametric curves $(t^{2} + 2 t, - 3 t^{2} + 5 t)$ $(t^2+2t,-3t^2+5t)$ and $(- 2 t^{2} + 4 t, t^{2} + 2 t)$ $(-2t^2+4t,t^2+2t)$ ?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you determine the intersection between parametric curves (t2+2t,−3t2+5t)(t^2+2t,-3t^2+5t) and (−2t2+4t,t2+2t)(-2t^2+4t,t^2+2t)?

1 Answer

Explanation:

Related questions

Impact of this question

How do you determine the intersection between parametric curves $(t^{2} + 2 t, - 3 t^{2} + 5 t)$ $(t^2+2t,-3t^2+5t)$ and $(- 2 t^{2} + 4 t, t^{2} + 2 t)$ $(-2t^2+4t,t^2+2t)$ ?