Let the system of equations : 2 x + 3 y + 5 z = 9 7 x + 3 y − 2 z = 8 12 x + 3 y − ( 4 + λ ) z = 16 − μ have infinitely many solutions. Then the radius of the circle centred at ( λ , μ ) and touch

Question

Let the system of equations : ​ 2 x + 3 y + 5 z = 9 7 x + 3 y − 2 z = 8 12 x + 3 y − ( 4 + λ ) z = 16 − μ ​ have infinitely many solutions. Then the radius of the circle centred at ( λ , μ ) and touching the line 4 x = 3 y is

TestHub · Accepted Answer

​ ​ 2 7 12 ​ 3 3 3 ​ 5 − 2 − ( λ + 4 ) ​ ​ = 0 ⇒ 12 ( − 21 ) − 3 ( − 39 ) − ( λ + 4 ) ( − 15 ) = 0 ⇒ − 252 + 117 + 15 ( 1 + 4 ) = 0 ⇒ 15 λ + 177 − 252 = 0 ⇒ 15 λ − 75 = 0 ⇒ λ = 5 ​ 9 8 16 − μ ​ 3 3 3 ​ 5 − 2 − 9 ​ ​ = 0 ⇒ ​ 1 μ − 8 16 − μ ​ 0 0 3 ​ 7 7 − 9 ​ ​ = 0 ⇒ 7 − 7 ( μ − 8 ) = 0 ⇒ 1 − ( μ − 8 ) = 0 ⇒ μ = 9 ⇒ centre of circle ( 5 , 9 ) radius = length of ⊥ from centre ( 5 , 9 ) = ​ 5 20 − 27 ​ ​ = 5 7 ​ ​

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