Can the intersection points between a circle and a parabola be algebraicly defined? I end up with polynomial degree 6. Thank you.

Circle equation:(x-m)^2+(y-n)^2=r^2 Parabola equation: (x-h)^2=4*f*(y-k)

Answer 1

[Shaik mohammed ghouse basha]

Hello,

As for my interpreation of your question, you are asking for a generalised equation for x and y coordinates when a circle of equation intersects with parabola of equation

The point of intersection lies on both circle and parabola. Let the point be of form (h+t, k + (t^2/4f) ).

This point lies on circle so we can write,

Upon expanding this equation you get a polynomial of degree 4.

Solving these equation you get possible values of t.

Using these value you can find the intersection points by inserting values of t in

This approach reduces polynomial to be solved from 6 degree to 4 degree but general terms can be quite complex.

For a four degree equation of form

we have solutions:

You can find a general equation by solving that four degree polynomial and substituing in coordinates but they turn out to be very complex.