A circle has a center that falls on the line #y = 5/4x +5 # and passes through # ( 4 ,7 )# and #(2 ,5 )#. What is the equation of the circle?
1 Answer
Explanation:
The centre of the circle must lie on the perpendicular bisector of the line segment joining
Their midpoint is:
#((4+2)/2, (7+5)/2) = (3, 6)#
and the slope of the line segment joining them is:
#(5-7)/(2-4) = (-2)/(-2) = 1#
Hence the slope of the perpendicular bisector is:
#-1/color(blue)(1) = -1#
Hence the equation of the perpendicular bisector can be written:
#y - 6 = -1(x - 3)#
which simplifies to:
#y = 9-x#
This will intersect the given line when:
#9-x = y = 5/4x+5#
Multiply both ends by
#36-4x=5x+20#
Add
#16 = 9x#
Hence:
#x = 16/9#
and:
#y = 9-x = 9-16/9 = 65/9#
So the centre of the circle is at:
So its equation can be written in the form:
#(x-16/9)^2 + (y-65/9)^2 = r^2#
where the radius
Since this passes through
So we find:
#r^2 = (color(blue)(4)-16/9)^2 + (color(blue)(7)-65/9)^2#
#color(white)(r^2) = (20/9)^2 + (-2/9)^2#
#color(white)(r^2) = 404/81#
So the equation of the circle may be written:
#(x-16/9)^2 + (y-65/9)^2 = 404/81#
graph{(y-x-3)((x-4)^2+(y-7)^2-0.006)((x-2)^2+(y-5)^2-0.006)((x-16/9)^2 + (y-65/9)^2 - 404/81)((x-16/9)^2 + (y-65/9)^2 - 0.006)(y+x-9)(y-5/4x-5)=0 [-3.63, 6.37, 4.6, 9.6]}