Let C :x^2+y^2=4 and C^ ' :x^2+y^2-4 λx+9=0 be two circles. If the set of all values of so that the circles C and C^ ' intersect at two distinct points, is R- [a ,b ], then the point (8a+12 ,16b-20 )…

Question

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

Accepted Answer

Correct answer: D. (d) x^2+y^2=4C (0 ,0 ) r_1=2C^ ' (2 ,0 ) r_2= 4 ^2-9(r_1-r_2)<CC^ '<(r_1+r_2)(2- 4 ^2-9)< |2 |<2+ 4 ^2-94+4 ^2-9-4 4 ^2-9<4 ^2 True ∈R... (i) 4 ^2<4+4 ^2-9+4 4 ^2-95<4 4 ^2-9 and ^2 94 2516<4 ^2-9 ∈(- ,- 32) ∪ ( 32 , )( 16964< ^2) ∈(- ,- 138) ∪ ( 138 , ) ( ii ) from (i) and (ii) ∈ ∈(- ,- 138) ∪ ( 138 , ) ⇒R-(- 138 , 138) as per question a=- 138 and b= 138 ∴ required point is (-1 ,6 ) with satisfies option (d)

$Let C : x^{2} + y^{2} = 4 and C^{’} : x^{2} + y^{2} - 4 λx + 9 = 0 be two circles. If the set of all values of λ so that the circles C and C^{’} intersect at two distinct points, is R - [a, b], then the point (8 a + 12, 16 b - 20) lies on the curve :$

Solution

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