Q: Let S be the set of all possible values of parameter ‘a’ for which the points of intersection of the parabolas y^{2} = 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

y^{2} – 3ax + λ(x^{2} + ax + 5 – 2y) = 0

it will represents a circle if λ = 1

x^{2} + y^{2} – 2ax – 2y + 5 = 0

it represents a real circle if a^{2} + 1 – 5 > 0

⇒ a^{2} > 4

⇒ a ∈ (-∞, -2) ∪ (2, ∞).

Hence (A) and (D) are the correct answers.