A circle has its center at (2,2) and is tangent to both the x-axis and y-axis. A line tangent to this circle intersects the x-axis at (a,0) and the y-axis at (0,b). If the shaded area is equal to the area of the circle, then a + b = ? (EXACT ANSWER).

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Answer 1 · 2017-12-17T17:30:59

#a+b=(ab)/4+2#

A circle has its center at #(2,2)# and is tangent to both #x#-axis and #y#-axis, then its radius would be #2# and equation is

#(x-2)^2+(y-2)^2=4#

The circle appears as

graph{(x-2)^2+(y-2)^2=4 [-2.73, 7.27, -0.46, 4.54]}

As the liine tangent to this circle intersects the #x#-axis at #(a,0)# and the #y#-axis at #(0,b)#, theequaation of tangent is

#(x-0)/(y-b)=(a-0)/(0-b)#

or #x/(y-b)=-a/b#

or #bx=-ay+ab#

or #ay+bx-ab=0#

its distance from center #(2,,2)# is #2#

#(axx2+bxx2-ab)/sqrt(a^2+b^2)=2#

and squaring

#(2a+2b-ab)^2=4(a^2+b^2)#

or #4a^2+4b^2+a^2b^2-4a^2b-4ab^2+8ab=4a^2+4b^2#

or #a^2b^2-4a^2b-4ab^2+8ab=0#

as #ab!=0#, dividing by #ab# we get

#ab-4a-4b+8=0#

or #4a+4b=ab+8#

or #a+b=(ab)/4+2#