Two circles have the following equations #(x -1 )^2+(y -2 )^2= 64 # and #(x +3 )^2+(y -4 )^2= 9 #. Does one circle contain the other? If not, what is the greatest possible distance between a point on one circle and another point on the other?

1 Answer
Feb 25, 2018

#"one circle inside other"#

Explanation:

#"the equation of a circle in standard form is"#

#•color(white)(x)(x-a)^2+(y-b)^2=r^2#

#"where "(a,b)" are the coordinates of the centre and r is"#
#"the radius"#

#(x-1)^2+(y-2)^2=64" has centre "(1,2)" and "r=8#

#(x+3)^2+(y-4)^2=9" has centre "(-3,4),r=3#

#"what we have to do here is "color(blue)"compare ""the distance"#
#"( d) between the centres to the "color(blue)"sum/difference of radii"#

#• " if sum of radii">d" then circles overlap"#

#• " if sum of radii"< d" then no overlap"#

#• " if difference of radii">d" then 1 circle inside other"#

#"to calculate d use the "color(blue)"distance formula"#

#•color(white)(x)d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)#

#"let "(x_1,y_1)=(1,2)" and "(x_2,y_2)=(-3,4)#

#d=sqrt((-3-1)^2+(4-2)^2)=sqrt(16+4)=sqrt20~~4.47#

#"sum of radii "=8+3=11#

#"difference of radii "=8-3=5#

#"since diff. of radii">d" then 1 circle inside other"#
graph{((x-1)^2+(y-2)^2-64)((x+3)^2+(y-4)^2-9)=0 [-40, 40, -20, 20]}