If is a point on the circle whose center is on the X axis and which touches the line at , then the greatest value of is
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Since the line x + y = 0 is a tangent, the line normal to the tangent (this line will pass through the center of the circle) will have equation of y = x + C
to determine C , notices that (2,-2) is a point of that normal. Thus
− 2 = 2 + C
C = − 4
The centre of the circle is on the X-axis, so to figure out the coordinate of the centre:
y = x − 4
0 = x − 4
x = 4
Thus the centre of the circle is located at ( 4 , 0 ) .
To complete the equation of the circle, notice that (2,-2) is a point on the circle.
k 2 ( x − 4 ) 2 + k 2 y 2 = 1
k 2 ( − 2 ) 2 + k 2 ( − 2 ) 2 = 1
k 2 4 + k 2 4 = 1
k 2 = 8
k = 2 2
As ( α , β ) is on the circle, maximum value of α occurs at ( α , 0 ) which gives
( 2 2 ) 2 ( α − 4 ) 2 + ( 2 2 ) 2 ( 0 ) 2 = 1
( 2 2 ) 2 ( α − 4 ) 2 = 1
= ( α − 4 ) 2 ( 2 2 ) 2
α = 4 ± 2 2
These points are where the circle intersects with the x-axis. We take the larger value, α = 4 + 2 2 in this case for the maximum.