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In this video you will learn how to find the intersection of a line and a circle algebraically.

Transcript

Finding the intersection of a Circle and a line algebraically
We will solve the system. We have a circle and a line. The may not cross and you wouldn't have a solution. They might touch once, or they may touch twice. If you have a circle and a line my go through the line twice and have two points of intersection. It can't have more than two points of intersection.
How do we do this
First we will solve the linear equation for a variable.
We will take the linear equation.
This is the one without any x squared in it and solve for one, whichever one is easier.
Step 2 we will substitute the value back into the linear equation to get the 2nd variable
Let's try this
First lets solve for x, because we can add y to both sides
x = y +2
Now lets plug in (y+2) for x
We now get (y+2)^2 + y^2 =34
We now need to get this as a quadratic in order to solve it.
Lets foil this out and we get
y^2 + 4y + 4 + y
I got that by foiling out (y+2)(y+2)
Let combine our like terms
We will get 2y^2 + 4y -30 =0
We now have the quadratic set to zero.
We can divide everything by 2 because it will make it easier.
We can now factor
What multiplies to 15 and has a difference of 2?
5 and 3 would work and it will be a +5 and a negative 3
What are the solutions?
Set these equal to zero, and (x+5) =0 equals -5
and (x-3) =0 equals 3
However I don't have a point.
I now have the x values, I now need to substitute the x value back into the linear function to get the y value.
What a minute, these are y values, sorry
I didn't solve correctly, these are y values
I will plug these in for y not for x
Lets plug in y as -5
x +5 = -3
So that point is negative 3 because that is my x coordinate ( -3,-5)
Now lets solve for y =3
x -3 =2
So x =5
The x value is 5 when the y value is 3
5,3)

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