The circles \(x^2 + y^2 = 4\) and \((x - 3)^2 + (y - 5)^2 = 25\) intersect in two points, say \(A\) and \(B\). Find the slope of line segment \(\overline{AB}\).
Thanks!
+36916 x^2 + y^2 -4 = 0 and (x-3)^2 + (y-5)^2 - 25 = 0 equate the two and expand and simplify to get
6x + 10y-13 = 0 this is a line
10 y = -6x + 13
y = -6/10 x + 13 slope = -6/10 = -3/5 (I used desmos to look this over graphically to verify this solution)
Just as Chris found !
+128472 x^2 + y^2 = 4
(x-3)^2 + (y -5)^2 = 25
Expand the second equation
x^2 - 6x + 9 + y^2 - 10y + 25 = 25 sub the first equation into this for x^2, y^2
4 - 6x + 9 - 10y + 25 = 25 simplify
13 - 6x - 10y = 0
10y = 13 - 6x
y = [ 13 - 6x ] / 10 sub this into the first equation fo y
x^2 + ( [ 13 - 6x ] / 10)^2 = 4 multiply through by 10^2 and simplify
100x^2 + 169 - 156x + 36x^2 = 400
136x^2 - 156x - 231 = 0
Solving this for x produces
x = ( 39 + 25√15) / 68 and x = ( 39 - 25√15) /68
And using a little technology [ because of the messy math ] when x holds the first value, y = [ 13 - 6x ] /10 =
(65 - 15√15) / 68
And when x holds the second value, y = [65 + 15√15) /68
So letting A = [ ( 39 - 25√15) /68 , ( 65 +15√15) /68 ] and B = [ ( 39 + 25√15) /68 , ( 65 - 15√15) /68
We can ignore the denominators when calculating the slope.....so we have
[ 65 - 15√15] - [ 65 + 15√15 ] -30√15 -3
_________________________ = ________ = ____
[39 + 25√15 ] - [ 39 - 25√15 ] 50√15 5
Thanks very much. Great explanation! I love this site and the things you guys do! I miss reading MAD Magazine Lol.
+36916 x^2 + y^2 -4 = 0 and (x-3)^2 + (y-5)^2 - 25 = 0 equate the two and expand and simplify to get
6x + 10y-13 = 0 this is a line
10 y = -6x + 13
y = -6/10 x + 13 slope = -6/10 = -3/5 (I used desmos to look this over graphically to verify this solution)
Just as Chris found !
