Q: Let S be the set of all possible values of parameter ‘a’ for which the points of intersection of the parabolas y2 = 3ax and $y = \frac{1}{2}(x^2 + ax + 5)$ y = are concyclic . Then S contains the interval(s)
(A) (– ∞ , – 2)
(B) (– 2, 0)
(C) (0, 2)
(D) (2, ∞)
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Ans: (A), (D)
Sol: Family of curves passing the points of intersection of two parabolas is
y2 – 3ax + λ(x2 + ax + 5 – 2y) = 0
it will represents a circle if λ = 1
x2 + y2 – 2ax – 2y + 5 = 0
it represents a real circle if a2 + 1 – 5 > 0
⇒ a2 > 4
⇒ a ∈ (-∞, -2) ∪ (2, ∞).
Hence (A) and (D) are the correct answers.