Hi
Can you please answer this question and show me the solution, cheers.

Show that the line 3x-4y+15=0 is a tangent to the circle x^2+y^2=9

Thanks

You need to prove that your two curves cross at only one point, that defines a tangent.

How would you do this, do you think?

I'm not sure. I don't really understand the concept of tangents and the unit circle.
I just need the solution to it then it will get easier.

I just told you what a tangent is, it's a line that crosses at only one point (not 2 or 0 points).

You don't have a unit circle, you have a circle of radius 3, so you don't need to worry about that.

1: find the point(s) of intersection between the two equations given
if step 1 finds more or less than 1 point of intersection then the line isn't tangent to the circle and you can quit here
2: find the slope of the circle at the point of intersection and the slope of the line
if the two slopes are equal then you've got a tangent line, otherwise you do not.

If you need help completing those steps please ask here.

I not entirely sure how to find the intersection of the first step.
I know simultaneous equations is invloved but I don't know what to do with the squared terms.

try solving the linear equation for one variable, substitute into the circle equation, solve for the remaining variable, substitute that value into the linear equation solved for 1 var to find the other variable.

Ok thanks
I eventually found the answer to be a perfect square which means that there can only be one point of intersection, in a tangent.
x=-9/5
y=12/5
What do I then do for the next step

Plug your answers into the circle equation. Do you get 9? Plug them into the linear equation. Do you get 0?

Yes it does.
Then what is the next step?

Next step? That's it.

OK. I guess I agree. I was wrong. You don't need to compare the slopes. The only way that a line can have a single point of intersection with a circle is if it is tangent to the circle at that point. I agree with galactus. Your math is done. You just need to explain what I just said.