Multiple choice

Construct a quadratic equation in $t$ such that $A.M.$ of its roots is $A$ and $G.M.$ is $G$.

  1. $t^{ 2 }-At+G^{ 2 }=0$.
  2. $t^{ 2 }-2At+G^{ 2 }=0$.
  3. $t^{ 2 }+At+G^{ 2 }=0$.
  4. $t^{ 2 }+2At+G^{ 2 }=0$.
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B Correct answer
Explanation

For a quadratic equation t^2 - (sum of roots)t + (product of roots) = 0, the sum of roots is 2 * A.M. = 2A. The product of roots is (G.M.)^2 = G^2. Thus, the equation is t^2 - 2At + G^2 = 0.

AI explanation

The arithmetic mean of the roots is A, so the sum of the roots alpha plus beta equals 2A. The geometric mean is G, so the product of the roots alpha times beta equals G squared. For a standard quadratic equation t squared minus (sum of roots)t plus (product of roots) equals 0, we substitute these values to get t squared minus 2At plus G squared equals 0.

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