How can you check if a line and a circle intersect?

Answer 1

Okay, so you simply plug the equation of the line into the equation of the circle.

(The reason you do this is because whenever they intersect, their x and y values are equal)

I'll give an example.

Say, for instance, you want to know if y=x+1 (the line) intersects with x2+y2=4 (the circle)

Plug in the x+1 for the y in the circle.

So you now have:

x2+(x+1)2=4

(Make sure that you include the parenthesis!)

x2+(x+1)(x+1)=4

x2+x2+2x+1=4

2x2+2x+1=4

2x2+2x-3=0

Then you will use part of the quadratic formula, which is:

x=(-b±√(b2-4ac))/2a

For this problem, you're interested in what the value under the radical sign is.

(b2-4ac)

Plug in the numbers

22-(4)(2)(-3)

4-(-24)

4+24

28

When you plug in the numbers for b2-4ac (known as the discriminant)

If the final result is:

Greater than 0 - you have 2 intersection points

Equal to 0 - you have 1 intersection point

Less than 0 - they do not intersect

So, for your original question, if the discriminant ends up being equal to or greater than 0, then the line and the circle intersect.

I hope that made sense.