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in Mathematics by (76.3k points)

Statement-1 : Though (λ, λ + 1) there can’t be more than one normal to the parabola y2 = 4x, if λ < 2. 

Statement-2 : The point (λ, λ + 1) lies outside the parabola for all λ ≠ 1. 

(A) Statement – 1 is True, Statement – 2 is True; Statement – 2 is a correct explanation for Statement – 1. 

(B) Statement – 1 is True, Statement – 2 is True; Statement – 2 is NOT a correct explanation for Statement – 1. 

(C) Statement – 1 is True, Statement – 2 is False. 

(D) Statement – 1 is False, Statement – 2 is True. 

1 Answer

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Best answer

Correct option: (b)

Explanation:

Any normal to y2 = 4x is  

Y + tx = 2t + t3 

If this passes through (λ, λ + 1), we get 

λ + 1 + λ = 2t + t

⇒ t3 + t(2 - λ) - λ - 1 = 0 = f(t) (say) 

If λ < 2, then f′(t) = 3t2 + (2 - λ) > 0 

⇒ f(t) = 0 will have only one real root. So A is true. 

Statement 2 is also true b′ coz (λ + 1)2 > 4λ is true ∀ λ ≠ 1. 

The statement is true but does not follow true statement-2. 

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