What is the difference between the maximum and the minimum value of x/y for which (x − 2)^2 = 9 and (y − 3)^2 = 25?

(A) −15/8
(B) 3/4
(C) 9/8
(D) 19/8
(E) 25/8
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Given data : \((x − 2)^2 = 9\) and \((y − 3)^2 = 25\)

To find the difference between the maximum and minimum values of \frac{x}{y}

\((x − 2)^2 = 9 -> (x-2) = 3\) or \((x-2) = -3\)
Maximum value of x : 5
Minimum value of x : -1

\((y − 3)^2 = 25 -> (y-3) = 5\) or \((y-3) = -5\)
Maximum value of y : 8
Minimum value of y : -2

Maximum value of \(\frac{x}{y} = \frac{5}{8}\), Minimum value of \(\frac{x}{y} = \frac{5}{-2}\)

Difference is \(\frac{5}{8} - (\frac{-5}{2}) = \frac{25}{8}\)(Option E)
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Damn, I have to proof read my solution.
Thanks a ton. Made the necessary change.
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If you solve respective equations for x and y, you will get as below.
X = 5 and -1
Y = 8 and -2
Just arrange all four combinations of x and y for our understanding of which is least and which is highest.

5/8, 5/-2, -1/8, -1/-2

Lowest (Most negative) is 5/-2 i.e. -2/5 and Highest is 5/8
So, 5/8 - ( -5/2) = 25/8
Hence answer is E.

Please give Kudos

(x − 2)^2 = 9
x - 2 = +/- 3
x = 5, -1

and

(y − 3)^2 = 25
y - 3 = +/- 5
y = 8, -2

So, min. value of x/y = min. { -5/2, -1/8} = -5/2
max. value of x/y = max. {5/8, 1/2} = 5/8

Difference between the two values = 5/8 - (-5/2) = 5/8 + 5/2 = 5/8 (1+4) = 25/8

Answer : (E) 25/8
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