Find the shortest distance from the line 3x + 4y = 25 to the circle x^2 + y^2 = 6x - 8y.
+128475 A graph might help to see what we have
Rearrange the equation of the circle 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
We cab write the equation of the given line as 3x + 4y - 25 = 0
Using the formula for the distance between a point ( the center of the circle ) and the given line we have that
l 3(3) + 4(-4) - 25 l / sqrt ( 3^2 + (4)^2) =
l -32 l / sqrt (25) =
32 / 5
Since rhe radius of the circle is 5.......then the distance between the circle and the line =
32/5 - 5 =
32/5 - 25/5 =
7 / 5 =
1.4 units
