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The equations $3 x - 7 y + k = 0$ and $12 x - ℓ y + 36 = 0$ have infinitely many solutions if

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a

$ℓ = 28, k \neq 9$

b

$ℓ \neq 28, k = 9$

c

$ℓ \neq 28, k \neq 9$

d

$ℓ = 28, k = 9$

answer is B.

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Detailed Solution

$3 x - 7 y + k = 0, 12 x - ℓ y + 36 = 0 h a v e i n f i n i t e l y m a n y s o l u t i o n s \Rightarrow \frac{3}{12} = \frac{- 7}{- ℓ} = \frac{k}{36} \Rightarrow ℓ = 28, k = 9 [f o r m u l a \frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}]$

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The equations 3x−7y+k=0 and 12x−ℓy+36=0 have infinitely many solutions if