H.M. between the roots of the equation ${x^2} - 10x + 11 = 0$ is :
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H.M. between the roots of the equation ${x^2} - 10x + 11 = 0$ is :
For x^2 - 10x + 11 = 0, roots alpha and beta have sum 10 and product 11. Harmonic mean HM = 2 * alpha * beta / (alpha + beta) = 2 * 11 / 10 = 22/10 = 11/5.
Let the roots of the quadratic equation be p and q, so their sum p + q = 10 and product pq = 11. The formula for the harmonic mean is 2pq / (p + q). Substituting the values gives 2(11) / 10, which equals 11/5.