The condition that the roots of the equation (a - x)(b - x) = c are in the ratio p : q is:
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(ab - c)(p + q)2 = pq(a + b)2
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(ab - c)(p + q)2 - pq (a2 + b2)2 = 0
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(ab - c)(p2 + q2) = pq(a + b)2
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None of these
Equation: x^2 - (a+b)x + ab - c = 0. Roots r1, r2. r1/r2 = p/q. r1+r2 = a+b, r1*r2 = ab-c. (r1+r2)^2 / (r1*r2) = (a+b)^2 / (ab-c). (p+q)^2 / pq = (a+b)^2 / (ab-c). Thus (ab-c)(p+q)^2 = pq(a+b)^2.
Expanding the given equation (a - x)(b - x) = c gives x^2 - (a + b)x + (ab - c) = 0. If the roots are in the ratio p : q, let them be pk and qk; their sum is pk + qk = k(p + q) = a + b, and their product is pk * qk = pqk^2 = ab - c. From the sum, k = (a + b) / (p + q), and substituting k^2 into the product equation gives pq * ((a + b)^2 / (p + q)^2) = ab - c. Rearranging this yields the required condition: (ab - c)(p + q)^2 = pq(a + b)^2.