Let the centre of a circle, passing through the points ( 0 , 0 ) , ( 1 , 0 ) and touching the circle x 2 + y 2 = 9 , be ( h , k ) . Then for all possible values of the coordinates of the centre ( h ,

Question

Let the centre of a circle, passing through the points ( 0 , 0 ) , ( 1 , 0 ) and touching the circle x 2 + y 2 = 9 , be ( h , k ) . Then for all possible values of the coordinates of the centre ( h , k ) , 4 ( h 2 + k 2 ) is equal to_________

TestHub · Accepted Answer

​ ( x − h ) 2 + ( y − k ) 2 = h 2 + k 2 x 2 + y 2 − 2 hx − 2 ky = 0 ∵ passes through ( 1 , 0 ) ⇒ 1 + 0 − 2 h = 0 ⇒ h = 1/2 ∵ OC = 2 OP ​ ( 2 1 ​ ) 2 + k 2 ​ = 2 3 ​ ​ ​ 4 1 ​ + k 2 = 4 9 ​ k 2 = 2 k = ± 2 ​ ​ ∴ Possible coordinate of ​ c ( h , k ) ( 2 1 ​ , 2 ​ ) ( 2 1 ​ , − 2 ​ ) 4 ( h 2 + k 2 ) = 4 ( 4 1 ​ + 2 ) = 4 ( 4 9 ​ ) = 9 ​

Mathematics - Circle Question with Solution | TestHub

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