SOLUTION: Hello I'm really stuck on this question and would be very grateful if you could help me out. QUESTION:Find the points of intersection of the line y=5x+13 with the circle {{{

Algebra ->  Graphs -> SOLUTION: Hello I'm really stuck on this question and would be very grateful if you could help me out. QUESTION:Find the points of intersection of the line y=5x+13 with the circle {{{       Log On


   

Question 291123: Hello
I'm really stuck on this question and would be very grateful if you could help me out.
QUESTION:Find the points of intersection of the line y=5x+13 with the circle +x%5E2%2By%5E2=13+
I'm not really sure how to approanch this question but have been attempting to solve it almost like a simultaneous equation by substitution:
+x%5E2%2B%285x%2B13%29%5E2=13+
This eventually gets me down to the quadratic:
+26x%5E2%2B130x%2B156=0+
simplified to +13x%5E2%2B65x%2B78=0+
This is all probably wrong but even if I am right I don't know where to go from here...
Please help!
Thanks ever so much,
From Katie
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
You're on the right track. I'll start where you left off.


Let's solve +13x%5E2%2B65x%2B78=0+ by use of the quadratic formula.

Solved by pluggable solver: Quadratic Formula
Let's use the quadratic formula to solve for x:


Starting with the general quadratic


ax%5E2%2Bbx%2Bc=0


the general solution using the quadratic equation is:


x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29




So lets solve 13%2Ax%5E2%2B65%2Ax%2B78=0 ( notice a=13, b=65, and c=78)





x+=+%28-65+%2B-+sqrt%28+%2865%29%5E2-4%2A13%2A78+%29%29%2F%282%2A13%29 Plug in a=13, b=65, and c=78




x+=+%28-65+%2B-+sqrt%28+4225-4%2A13%2A78+%29%29%2F%282%2A13%29 Square 65 to get 4225




x+=+%28-65+%2B-+sqrt%28+4225%2B-4056+%29%29%2F%282%2A13%29 Multiply -4%2A78%2A13 to get -4056




x+=+%28-65+%2B-+sqrt%28+169+%29%29%2F%282%2A13%29 Combine like terms in the radicand (everything under the square root)




x+=+%28-65+%2B-+13%29%2F%282%2A13%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)




x+=+%28-65+%2B-+13%29%2F26 Multiply 2 and 13 to get 26


So now the expression breaks down into two parts


x+=+%28-65+%2B+13%29%2F26 or x+=+%28-65+-+13%29%2F26


Lets look at the first part:


x=%28-65+%2B+13%29%2F26


x=-52%2F26 Add the terms in the numerator

x=-2 Divide


So one answer is

x=-2




Now lets look at the second part:


x=%28-65+-+13%29%2F26


x=-78%2F26 Subtract the terms in the numerator

x=-3 Divide


So another answer is

x=-3


So our solutions are:

x=-2 or x=-3





Now plug each of these solutions into y=5x%2B13 to find the corresponding 'y' coordinates of the intersections.

So when x=-2, then y=5x%2B13=5%28-2%29%2B13=-10%2B13=3 giving us the point (-2,3)


Also, when x=-3, then y=5x%2B13=5%28-3%29%2B13=-15%2B13=-2 giving us the point (-3,-2)


So the two points of intersection are (-2,3) and (-3,-2)