Two circles have the following equations #(x +5 )^2+(y -2 )^2= 36 # and #(x -1 )^2+(y -1 )^2= 81 #. 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. 21.08
Explanation:
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 coords of centre and r , the radius.Both equations here are in this form and so we can extract centres and radii by comparison with the standard equation.
# (x+5)^2+(y-2)^2=36" has centre (-5,2) and r = 6"#
# (x-1)^2+(y-1)^2=81" has centre (1,1) and r = 9"# Calculate the distance between centres using 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 the coords of 2 points "# let
# (x_1,y_1)=(-5,2)" and " (x_2,y_2)=(1,1)#
#rArr d=sqrt((1+5)^2+(1-2)^2)=sqrt(36+1)=sqrt37 ≈ 6.08# Now sum of radii = 6 + 9 = 15
Since sum of radii > d , then circles overlap.
greatest distance = d + sum of radii = 6.08 + 15 = 21.08
graph{(y^2-4y+x^2+10x-7)(y^2-2y+x^2-2x-79)=0 [-40, 40, -20, 20]}
