Multiple choice

Let x$^2$ - 2ax + b$^2$ = 0 and x$^2$ - 2bx + a$^2$ = 0 be two equations. Then the AM of the roots of the first equation is

  1. AM of the roots of second

  2. GM of the roots of second

  3. square root of the GM of the roots of second

  4. None of these
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B Correct answer
AI explanation

Let the roots of the first equation be r1 and r2; their arithmetic mean (AM) is (r1 + r2)/2. Using the sum of roots formula on x^2 - 2ax + b^2 = 0 gives r1 + r2 = 2a, so the AM of these roots is 2a/2, which equals a. For the second equation x^2 - 2bx + a^2 = 0 with roots r3 and r4, the product of the roots is r3 r4 = a^2, and the geometric mean (GM) is the square root of this product, meaning the GM equals a.

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