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).

Answer 1

#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#
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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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