Consider two circles C 1 : x 2 + y 2 = 25 and C 2 : ( x - α ) 2 + y 2 = 16 , where α ∈ ( 5 , 9 ) . Let the angle between the two radii (one to each circle) drawn from one of the intersection points of

Question

Consider two circles C 1 : x 2 + y 2 = 25 and C 2 : ( x - α ) 2 + y 2 = 16 , where α ∈ ( 5 , 9 ) . Let the angle between the two radii (one to each circle) drawn from one of the intersection points of C 1 and C 2 be sin - 1 63 8 . If the length of common chord of C 1 and C 2 is β , then the value of ( α β ) 2 equals _________.

TestHub · Accepted Answer

Given, C 1 : x 2 + y 2 = 25 , C 2 : ( x - α ) 2 + y 2 = 16 and 5 < α < 9 Now, plotting the diagram we get, Also, given sin θ = 63 8 Now, from the above diagram, finding the area of Δ OAP we get, Δ OAP = 1 2 × α β 2 = 1 2 × 5 × 4 sin θ ⇒ α β = 40 × 63 8 ⇒ α β = 5 × 63 ⇒ ( α β ) 2 = 25 × 63 = 1575

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