Let the point P of the focal chord PQ of the parabola y 2 = 16 x be ( 1 , − 4 ) . If the focus of the parabola divides the chord PQ in the ratio m : n , gcd ( m , n ) = 1 , then m 2 + n 2 is equal to

Question

Let the point P of the focal chord PQ of the parabola y 2 = 16 x be ( 1 , − 4 ) . If the focus of the parabola divides the chord PQ in the ratio m : n , gcd ( m , n ) = 1 , then m 2 + n 2 is equal to :

TestHub · Accepted Answer

y 2 = 16 x ; a = 4 focus S ≡ ( 4 , 0 ) ​ 2 at 1 ​ = − 4 ⇒ 2 ( 4 ) t 1 ​ = − 4 ⇒ t 1 ​ = − 2 1 ​ ∵ t 1 ​ t 2 ​ = − 1 ​ ​ ⇒ t 2 ​ = 2 ∴ Q ( at 2 2 ​ , 2 at 2 ​ ) = ( 16 , 16 ) ​ Let, S divides PQ internally in λ : 1 ratio ​ ∴ λ + 1 16 λ − 4 ​ = 0 λ = 4 1 ​ = n m ​ ∴ m 2 + n 2 = 1 + 16 = 17 ​

Mathematics - Parabola Question with Solution | TestHub

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