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).

1 Answer
Dec 17, 2017

a+b=(ab)/4+2

Explanation:

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