Two circles have the following equations: #(x -6 )^2+(y -4 )^2= 64 # and #(x +6 )^2+(y -9 )^2= 49 #. 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?
2 Answers
The circles overlap and the greatest distance is
Explanation:
The distance between the centers of the circles is
The sum of the radii is
As
The distance between the centers is
The circles overlap.
The greatest distance is
graph{((x-6)^2+(y-4)^2-64)((x+6)^2+(y-9)^2-49)=0 [-28.86, 28.85, -10.16, 18.7]}
Explanation:
#"what we have to do here is compare the distance (d )"#
#"between the centres of the circles to the "#
#color(blue)"sum/difference of the radii"#
#• " if difference of the radii > d then one circle is"#
#"contained within the other"#
#• " if sum of radii > d then circles overlap"#
#• " if sum of radii < d then no overlap"#
#• " the equation of a circle in standard form is"#
#color(red)(bar(ul(|color(white)(2/2)color(black)((x-a)^2+(y-b)^2=r^2)color(white)(2/2)|)))#
where ( a , b ) are the coordinates of the centre and r the radius.
#(x-6)^2+(y-4)^2=64to(6,4),r=8#
#(x+6)^2+(y-9)^2to(-6,9),r=7#
#"to calculate d use the "color(blue)"distance formula"#
#color(red)(bar(ul(|color(white)(2/2)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)|)))#
#(x_1,y_1)=(6,4),(x_2,y_2)=(-6,9)#
#rArrd=sqrt((-6-6)^2+(9-4)^2)=sqrt(144+25)=13#
#• " difference of radii "=8-7=1#
#"since difference of radii < d then circle not "#
#"contained in the other"#
#"sum of radii "=8+7=15#
#"since sum of radii > d then circles overlap"#
graph{(y^2-8y+x^2-12x-12)(y^2-18y+x^2+12x+68)=0 [-40, 40, -20, 20]}