Bunuel wrote:
In the above figure, the area of circle A is 144π and the area of circle B is 169π. If point x (not shown above) lies on circle A and point y (not shown above) lies on circle B, what is the range of the possible lengths of line xY.
A) 0 to 169π^2
B) 0 to 144
C) 0 to 25
D) 0 to 50
E) 5 to 144
Source: Platinum GMATKudos for a correct solution.Attachment:
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Platinum GMAT Official Solution:The shortest distance of line xy will occur when x and y are at the same point (i.e., the point where the two circles come together). In this instance, the line xy will be 0 units long.
In order to determine the longest possible distance for xy, we must first recall that the longest line across a circle is the circle's diameter. In other words, it is impossible to construct a line from one point on a circle to another point on the same circle that is longer than the circle's diameter.
The longest distance of line xy will occur when x and y are at exact opposite sides of the two circles. In other words, when x is at the far left of A and y is at the far right of B. More technically, line xy will be the combined diameter of circles A and B. This makes sense given that the diameter of a circle is the longest possible line from one point on the circle to another point on the same circle.
Since the length of xy is the length of the diameter of A plus the diameter of B, we need to find the diameter of each circle.
Area of A = 144π = πr^2
rA = 12 = radius of circle A
dA = 2(12) = 24 = diameter of circle A
Area of B = 169π = πr^2
rB = 13 = radius of circle B
dB = 2(13) = 26 = diameter of circle B
Maximum distance of xy = 24 + 26 = 50.
Answer: D.
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