Two circles have the following equations: #(x +2 )^2+(y -1 )^2= 16 # and #(x +4 )^2+(y +7 )^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
circles overlap.
Explanation:
What we have to do here is compare the distance ( d ) between the centres of the circles with the sum or difference of the radii.
There are 3 possible outcomes.• If sum of radii > d ,
#color(blue)"then circles overlap"# • If sum of radii < d ,
#color(blue)"then no overlap"# • If difference of radii > d
#color(red)"then 1 circle contains the other"# The standard form of the
#color(blue)"equation of a circle"# is
#color(red)(|bar(ul(color(white)(a/a)color(black)((x-a)^2+(y-b)^2=r^2)color(white)(a/a)|)))#
where (a ,b) are the coordinates of the centre and r, the radius.
#(x+2)^2+(y-1)^2=16rArr" centre"=(-2,1),r=4#
#(x+4)^2+(y+7)^2=25rArr"centre"=(-4,-7),r=5# To calculate d, use the
#color(blue)"distance formula"#
#color(red)(|bar(ul(color(white)(a/a)color(black)(d=sqrt((x_2-x_1)^2+(y_2-y_1)^2))color(white)(a/a)|)))#
where# (x_1,y_1)" and " (x_2,y_2)" are 2 coordinate points"# The 2 points here are (-2 ,1) and (-4 ,-7) the centres of the circles.
#d=sqrt((-4+2)^2+(-7-1)^2)=sqrt(4+64)=sqrt68≈8.246# sum of radii = 4 + 5 = 9
difference of radii = 5 - 4 = 1
Since sum of radii > d , then circles overlap
graph{(y^2-2y+x^2+4x-11)(y^2+14y+x^2+8x+0)=0 [-40, 40, -20, 20]}