This question is from CAT 2022 slot 3 . I’m not sure how to check which value is not possible.?

Question

If c=16x/y +49y/x  for some non-zero real numbers x and y , then c cannot take the value
A. -70
B. 60
C. -50
D. -60

V Ritik · Accepted Answer

This is a standard expression type where checking limits through substitution works well.

Start with c = 16x/y + 49y/x
Let a = x/y and b = y/x so that ab = 1
Rewrite the expression as c = 16a + 49b
Try values like a = 1, b = 1 gives c = 65; a = &ndash;1, b = &ndash;1 gives c = &ndash;65
Try a = &ndash;2, b = &ndash;0.5 gives c = &ndash;56.5

No matter what values you try under ab = 1, c never reaches &ndash;70. So the value that c cannot take is &ndash;70. The correct answer is Option A.

D. Aarav · Answer

This expression is of the form c = 16x/y + 49y/x. The trick is to rewrite it using a substitution: let a = x/y and b = y/x. Since a and b are reciprocals, ab = 1. So the expression becomes c = 16a + 49b. Now, apply AM &ge; GM: (16a + 49b)/2 &ge; &radic;(16a &times; 49b) = &radic;(784ab) = &radic;784 = 28. That means the minimum value of c is at least 28 if both terms are positive. But x and y can also be negative, so a and b could be negative as well. Try extreme values like a = &ndash;1 and b = &ndash;1, and you get c = &ndash;16 &ndash; 49 = &ndash;65. Even that doesn&rsquo;t go beyond &ndash;70.
So when you test different values of a and b under the constraint ab = 1, c never touches &ndash;70. That&rsquo;s the one value c cannot take. The answer is Option A &ndash;70.