The combined equation of the straight lines of the form y = kx+1 (where k is an integer) such that the point of intersection of each with the line 3x+4y=9 has an integer as its x- coordinate is

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(y+x+1) (y+2x-1)=0

(y+x-1)(y+2x+1)=0

(y+x+1)(y+2x+1)=0

(y+x-1)(y+2x-1)=0

answer is D.

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Detailed Solution

$\begin{array}{l} G i v e n l i n e s y = k x + 1 - - (1) \\ 3 x + 4 y = 9 - - - (2) \\ p u t (1) i n (2) \\ \Rightarrow 3 x + 4 (k x + 1) = 9 \\ \Rightarrow 3 x + 4 k x + 4 = 9 \\ \Rightarrow x (3 + 4 k) = 5 \\ \Rightarrow x = \frac{5}{3 + 4 k} \\ f o r x t o b e a n int e g e r; 3 + 4 k s h o u l d d i v i d e' 5' \\ ∴ D i v i s o l s o f 5 a r e {1, 5, - 1, - 5} \\ I f 3 + 4 k = 1 \Rightarrow k = \frac{- 1}{2} \\ I f 3 + 4 k = 5 \Rightarrow k = \frac{1}{2} \\ I f 3 + 4 k = - 1 \Rightarrow k = - 1 \\ I f 3 + 4 k = - 5 \Rightarrow k = - 2 \\ ∵ k i s a n int e g e r \Rightarrow k = - 1, - 2 \\ ∴ f r o m (1) t h e l i n e s y = - x + 1 \Rightarrow x + y - 1 = 0 \\ a n d y = - 2 x + 1 = 2 x + y - 1 = 0 \\ c o m b a i n e d E q u a t i o n i s (x + y - 1) (2 x + y - 1) = 0 \end{array}$