Let C : x 2 + y 2 = 4 and C ' : x 2 + y 2 − 4 λ x + 9 = 0 be two circles. If the set of all values of λ so that the circles C and C ' intersect at two distinct points, is R − a , b , then the point 8 a + 12 , 16 b − 20 lies on the curve:

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Correct answer: D. 6 x 2 + y 2 = 42

Given: x 2 + y 2 = 4 So, centre and radius of C are 0 , 0 and r 1 = 2 respectively. Also, C ' : x 2 + y 2 − 4 λ x + 9 = 0 So, centre and radius of C ' are 2 λ , 0 and r 2 = 4 λ 2 − 9 respectively. ⇒ r 1 − r 2 < C C ' < r 1 + r 2 ⇒ 2 − 4 λ 2 − 9 < 2 λ < 2 + 4 λ 2 − 9 ⇒ 4 + 4 λ 2 − 9 − 4 4 λ 2 − 9 < 4 λ 2 ⇒ λ ∈ R . . . i Also, 4 λ 2 < 4 + 4 λ 2 − 9 + 4 4 λ 2 − 9 ⇒ 5 < 4 4 λ 2 − 9 λ 2 ≥ 9 4 ⇒ λ ∈ − ∞ , − 3 2 ∪ 3 2 , ∞ ⇒ 25 16 < 4 λ 2 − 9 ⇒ 169 64 < λ 2 ⇒ 169 64 < λ 2 ⇒ λ ∈ − ∞ , − 13 8 ∪ 13 8 , ∞ . . . i i Using i and

Let C : x 2 + y 2 = 4 and C ' : x 2 + y 2 − 4 λ x + 9 = 0 be two circles. If the set of all values of λ so that the circles C and C ' intersect at two distinct points, is R − a , b , then the point 8 a + 12 , 16 b − 20 lies on the curve:

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