Find the shortest distance from the line 3x + 4y = 25 to the circle x^2 + y^2 = 6x - 8y.
+128475 First.....we need to find the circle center
Rewrite as
x^2 - 6x + y^2 + 8y = 0 complete the square on x and y
x^2 - 6x + 9 + y^2 + 8y + 16 = 9 + 16
(x - 3)^2 + ( y + 4)^2 = 25
This is a circle with a center of ( 3, -4) and a radius of 5
Now.....we can use a special formula that will tell us how far the center of the circle is from our given line
We are using the equation of the line in standard form and filling in x and y with the circle's center
The squre root in the denominator is just the coefficients of the line (each squared)
We have that
l 3(3) + 4(-4) - 25 l l -32 l 32
________________ = ________ = __
sqrt [ 3^2 + 4^2 ] sqrt (25) 5
Subtract the radius of the circle from this
32 / 5 - 5 =
32 /5 - 25 /5 =
7/5 = 1.4 units = the shortest distance between the given line and the circle
