Assertion (A) : All chords of the curve $4 x^{2} + y^{2} - x + 4 y = 0,$ which subtends right angle at the origin passes through the point $(\frac{1}{5}, - \frac{4}{5})$
Reason( R): Chords of any curve, substending right angle at origin passes through a fixed point

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Both A & R are true R is not correct explanation of A

A is false R is true

A is true R is false

Both A & R are true R is correct explanation of A

answer is C.

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Detailed Solution

$\begin{array}{l} G i v e n E q u a t i o n 4 x^{2} + y^{2} - x + 4 y = 0 - - (1) \\ L e t e q u a t i o n o f c h o r d y = m x + c \Rightarrow \frac{y - m x}{c} = 1 - - (2) \\ H o m o g e n i z i n g (1) w i t h (2) \\ 4 x^{2} + y^{2} - x (\frac{y - m x}{c}) + 4 y (\frac{y - m x}{c}) = 0 \\ \Rightarrow 4 x^{2} c + y^{2} c - x y + m x^{2} + 4 y^{2} - 4 m x y = 0 \\ T h e l i n e s a r e ⊥ r i f f \\ c o . e f f x^{2} + c o . e f f y^{2} = 0 \\ \Rightarrow 4 c + 1 + m + 4 = 0 \\ \Rightarrow m = - 4 c - 5 \\ ∴ t h e c h o r d i s y = (- 4 c - 5) x + c \\ \Rightarrow 4 c x + 5 x + y - c = 0 \\ \Rightarrow (5 x + y) + (4 x - 1) c = 0 \\ I f i s o f t h e f o r m L_{1} + λ L_{2} = 0 \end{array}$

$\begin{array}{l} L_{1} = 5 x + y = 0 \\ L_{2} = 4 x - 1 = 0 \Rightarrow x = \frac{1}{4} \\ y = - 5 x \Rightarrow \frac{- 5}{4} \\ ∴ p o int i s (\frac{1}{4}, \frac{- 5}{4}) \\ \Rightarrow 5 c x + 4 x + y - c = 0 \\ \Rightarrow (4 x + y) + c (5 x - 1) = 0 \\ I f i s o f t h e f o r m L_{1} + λ L_{2} = 0 \\ 4 x + y = 0 - - (3) \\ 5 x - 1 = 0 \Rightarrow x = \frac{1}{5} \\ f r o m (3) y = \frac{- 4}{5} \\ ∴ p o int (\frac{1}{5}, \frac{- 4}{5}) \\ T r u e \\ Re a s o n (R) : F a l s e \end{array}$