Question 1207163
circle {{{a}}}:
{{{r=10in}}}

circle {{{b}}}:
{{{r=16in}}}

distance between centers={{{22in}}}

if we place the center of the circle a on {{{x}}}-ais, at point ({{{-11}}},{{{0}}}), than the center of the circle b will be at ({{{11}}},{{{0}}}) and the distance between them is {{{22in}}}


now we have equations:

circle {{{a}}}

{{{(x-(-11))^2+(y-0)^2=10^2 }}}

{{{(x+11)^2+y^2=100}}}


circle{{{ b}}}

{{{(x-11)^2+(y-0)^2=16^2 }}}

{{{(x-11)^2+y^2=256}}}


find intersection points by solving this system:

{{{(x-11)^2+y^2=256}}}
{{{(x+11)^2+y^2=100}}}
-----------------------------subtract

{{{(x-11)^2+y^2-(x+11)^2-y^2=256-100}}}

{{{(x-11)^2-(x+11)^2=156}}}

{{{x^2 - 22x + 121-(x^2 + 22x + 121)=156}}}

{{{x^2 - 22x + 121-x^2 -22x - 121=156}}}

 {{{- 22x  -22x =156}}}

{{{- 44x =156}}}

{{{x=156/-44}}}

{{{x=-39/11}}}



plug it in

{{{(-39/11+11)^2+y^2=100}}}

{{{6724/121+y^2=100}}}

{{{y^2=100-6724/121}}}

{{{y^2=5376/121}}}

{{{y=sqrt(5376/121)}}}

{{{y = (16 sqrt(21))/11 }}}or {{{y =- (16 sqrt(21))/11}}}


solutions:

{{{x = -39/11}}}, {{{y = (16sqrt(21))/11}}}

or

{{{x = -39/11}}}, {{{y = -(16sqrt(21))/11}}}


intersection points are

({{{-39/11}}}, {{{(16sqrt(21))/11}}}) and ({{{-39/11}}}, {{{-(16sqrt(21))/11}}}) 


distance between is: two times {{{y}}} value 

{{{d=2*(16sqrt(21))/11=13.33}}}


The length of {{{cd}}} is {{{13.33in}}}.


{{{ drawing( 600, 600, -30, 30, -30, 30,
circle(-11,0,.35),circle(11,0,.35),
circle(-11,0,10),circle(11,0,16), locate(-15,9.5,a), locate(21,12,b),
circle(-39/11,(16sqrt(21))/11,.35),circle(-39/11,-(16sqrt(21))/11,.35),
locate(-39/11,(16sqrt(21))/11,c),locate(-39/11,-(16sqrt(21))/11,d),
blue(line(-39/11,(16sqrt(21))/11,-39/11,-(16sqrt(21))/11)),
graph( 600, 600, -30, 30, -30, 30, 0)) }}}