Two circles have the following equations: #(x -8 )^2+(y -5 )^2= 64 # and #(x +4 )^2+(y +2 )^2= 25 #. 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
No overlap, greatest distance ≈ 26.89 units
Explanation:
What we have to do here is
#color(blue)"compare"# the distance (d) between the centres of the circles to the#color(blue)"sum/difference of the radii"# • If sum of radii > d , then circles overlap
• If sum of radii < d , then no overlap
• If diff. of radii > d , then 1 circle inside other
We require to find the centres and radii of the circles.
The standard form of the
#color(blue)"equation of a circle"# 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.
#rArr(x-8)^2+(y-5)^2=64to" centre "=(8,5),r=8#
#(x+4)^2+(y+2)^2=25to"centre "=(-4,-2),r=5# 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))color(white)(2/2)|)))#
where# (x_1,y_1),(x_2,y_2)" are 2 coordinate points"# let
# (x_1,y_1)=(8,5)" and " (x_2,y_2)=(-4,-2)#
#d=sqrt((-4-8)^2+(-2-5)^2)=sqrt193≈13.89# sum of radii = 8 + 5 = 13
and diff. of radii = 8 - 5 = 3
Since diff. of radii < d , then 1 circle NOT inside the other
Since sum of radii < d , then no overlap of circles
Greatest distance between 2 points = sum of radii + d
#=13+13.89=26.89" units (to 2 decimal places)"#
graph{(y^2-10y+x^2-16x+25)(y^2+4y+x^2+8x-5)=0 [-40, 40, -20, 20]}