Let \alpha and \beta be the roots of the equation x^{2} + qx - r = 0, where p eq 0. If p,q and r be the consecutive terms of a non constant G.P. and \frac{1}{\alpha} + \frac{1}{\beta} = \frac{3}{4}, then the value of (\alpha - \beta)^{2} is

Question

Let alpha and beta be the roots of the equationn x^2 + qx - r = 0, where p ≠ 0. If p,q and rn be the consecutive terms of a non constant G.P. andn (1 / alpha) + (1 / beta) = (3 / 4), then the valuen of (alpha - beta)^2 is

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Step 1: Identify the roots and relationshipsn nLet alpha and beta be the roots of the quadratic equationnn$$nnpx^2 + qx - r = 0.nn$$n nBy Vieta's formulas:nn$$nnalpha + beta = -(q / p) 1nn$$nn$$nnalpha beta = -(r / p) 2nn$$n nStep 2: Use the given conditionn nGiven:nn$$nn(1 / alpha) + (1 / beta) = (3 / 4).nn$$n nThis can be rewritten as:nn$$nn(alpha + beta / alpha beta) = (3 / 4).nn$$n nSubstituting from (1) and (2):nn$$nnfrac-(q / p)-(r / p) = (3 / 4)nn$$n n$$nn => (q / r) = (3 / 4). 3nn$$n nStep 3: Establish the G.P. relationshipn nSince p, q, r are consecutive terms of a non-constant geometric progression, let the common ratio be k:nn$$nn(q / p) = k, (r / q) = k.nn$$n nFrom (3):nn$$nn(r / q) = (4 / 3).nn$$n nHence:nn$$nnr = (4 / 3)q. 4nn$$n nStep 4: Express p in terms of qn nFrom (q / p) = k:nn$$nnp = (q / k). 5nn$$n nStep 5: Find (alpha - beta)^2n nWe use the identity:nn$$nn(alpha - beta)^2 = (alpha + beta)^2 - 4alphabeta.nn$$n nSubstituting from (1) and (2):nn$$nn(alpha - beta)^2 = (-(q / p))^2 - 4(-(r / p))nn$$n n$$nn= (q^2 / p^2) + (4r / p).nn$$n nStep 6: Substitute the value of rn nUsing r = (4 / 3)q:nn$$nn(alpha - beta)^2 = (q^2 / p^2) + (16q / 3p).nn$$n nStep 7: Express everything in terms of kn nSince p = (q / k):nn$$nn(alpha - beta)^2nn= fracq^2((q / k))^2nn+ frac16q3((q / k))nn$$n n$$nn= k^2 + (16k / 3).nn$$n nStep 8: Find the value of kn nFrom earlier:nn$$nn(q / p) = (4 / 3) => k = (4 / 3).nn$$n nSubstituting:nn$$nn(alpha - beta)^2nn= ((4 / 3))^2 + (16 / 3) * (4 / 3)nn= (16 / 9) + (64 / 9)nn= (80 / 9).nn$$nn

Explanation

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