Intersection of line and circle

[markraz]

Hi I need to find the the intersection(s) of a line and a cicle. I found a solution using Quadratic formula but how can this be done using matrix/linear algebra?? I cannot find any example and none of my books show this. Does anyone know how to do this? Thanks in advance

 

[yoscar04]

The equation of a circle is non linear but you want to invoke linear algebra to solve its intersection with a line. I am curious to see if any one from the forum can answer in an affirmative way to your question.
From Paul's online notes:
In this section we are going to be looking at non-linear systems of equations. A non-linear system of equations is a system in which at least one of the variables has an exponent other than 1 and/or there is a product of variables in one of the equations.

To solve these systems we will use either the substitution method or elimination method that we first looked at when we solved systems of linear equations. The main difference is that we may end up getting complex solutions in addition to real solutions. Just as we saw in solving systems of two equations the real solutions will represent the coordinates of the points where the graphs of the two functions intersect.

 

[yoscar04]

You could write your system of equations as:
[x-2 y-4][x]=0
[2 1 ][ y]=9
I apologize for the notation, but I meant a 2x2 matrix times a 2x1 column vector . Now you may try generalized Cramer's rule.
For more information look for: Nonlinear matrix algebra and engineering applications. Part 1 : Theory and linear form matrix. C. L. Wu and R. J. Adler. It looks a bit cumbersome though.

 

[Dr.Peterson]

Do you want to use linear algebra because you think it will be easier, or because you want to make it harder?

Also, are you assuming you are given the equations as in the example, or the center and radius, or what? I suppose you might be able to use the line through the circle's center perpendicular to the line as part of your work ...

 

[pka]

Hi I need to find the the intersection(s) of a line and a cicle. I found a solution using Quadratic formula but how can this be done using matrix/linear algebra??

We want to find the intersection of \(x^2+y^2-2x-4y=0~\&~y=-2x+9\)
The circle is \(x-1)^2+(y-2)^2=5\\(x-1)^2+(-2x+9-2)^2=5\\(x-1)^2+(-2x+7)^2=5\\x^2-2x+1+4x^2-28x+49=5\\5x^2-30x+50=5\\x^2-6x+9=0\\(x-3)^2=0\\x=3\)

 

[markraz]

Do you want to use linear algebra because you think it will be easier, or because you want to make it harder?

Also, are you assuming you are given the equations as in the example, or the center and radius, or what? I suppose you might be able to use the line through the circle's center perpendicular to the line as part of your work ...

Hi, Thanks for the reply.
I am writing a computer game and I am trying to make sure I don't use any division to keep the processing as fast as possible

Thanks

 

[markraz]

Thanks for the help everyone