Find the points of intersection between x^2+y^2=45 and -3x+y=15

Step-by-step

Given:

Circle C1: x^2+y^2 = 45

Line L1: -3x+y=15

Need to find the points of intersection.

Solution:

basically we need to solve for the roots of equations C1 and L1.

Here, we can use substitution of L1 into C1.

Rewrite L1 as : y=3x+15

substitute into C1:

x^2+(3x+15)^2 = 45

Expand

x^2 + 9x^2+90x+225 = 45

Rearrange terms:

10x^2+90x+180 = 0

Simplify

x^2+9x+18 = 0

Factor

(x+6)(x+3) = 0

so

x=-6 or x=-3

Back-substitute x into L1 to calculate y:

x=-6, y=3*x+15 = 3(-6)+15 = -3  => (-6,-3)

x=-3, y=3*x+15 = 3(-3) + 15 = 6 => (-3, 6)

Therefore the intersection points are (-6,-3) and (-3,6)

Check using equation C1:

(-6)^2+(-3)^2 = 36+9 = 45  ok

(-3)^2+(6)^2 = 9 + 36 = 45 ok

Check using equation L1:

Point (-6,-3) : y = 3x+15 = 3(-6) +15 = -3  ok

Point (-3,6) : y = 3x+15 = 3(-3)+15 = 6 ok.