Maximum And Minimum Of Roots

Real roots of a quadratic require a non-negative discriminant, the primary JEE Advanced condition for parametric root problems. For x^2 - (k - 3)x + k = 0, the discriminant is D = (k - 3)^2 - 4k. Expanding gives k^2 - 6k + 9 - 4k = k^2 - 10k + 9. Requiring D ≥ 0 means k^2 - 10k + 9 ≥ 0, which factors as (k - 1)(k - 9) ≥ 0. This product is non-negative when k ≤ 1 or k ≥ 9, the regions outside the roots of the auxiliary quadratic. Option 1 ≤ k ≤ 9 selects where the discriminant is negative, giving complex roots. Option k ≥ 9 only drops the lower branch. Option all real k ignores the constraint entirely. Hence k ≤ 1 or k ≥ 9. Plausibility check: at the boundary k = 1, D = (1 - 9)(1 - 1)? compute D = 1 - 10 + 9 = 0, giving equal real roots, confirming the endpoints belong to the solution set. The argument of a difference ratio measuring the angle a segment subtends at a moving point is one of the most powerful locus tools in the Argand plane, producing circles, arcs and lines from clean angle conditions. The angle-in-a-semicircle theorem, here giving a right angle on a circle with the segment as diameter, is the geometric backbone that converts an algebraic argument equation into a recognizable curve.