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Question

A system of linear equations is given as follows :
a1x+b1y+c1=0
a2x+b2y+c2=0
Both the lines are parallel only if,

A
a1a2=b1b2
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B
a1a2=b1b2c1c2
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C
a1a2=b1b2=c1c2
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D
None of these
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Solution

The correct option is B a1a2=b1b2c1c2
Letustaketwoequationsas,
a1x+b1y+c1=0........(i)anda2x+b2y+c2=0.......(ii).

IfthethelinesareparallelthentheslopesofthetwolinesshouldbeequalandtheinterceptsofthetwolinesshouldNOTbeequal.

Writingtheequationsoflinesinslopeinterceptformwehave,

(i)asy=a1b1x+(c1b1)and(ii)asy=a2b2x+(c2b2)

Nowifthelinesareparallelthentheslopesareequal.

i.ea1b1=a2b2a1a2=b1b2........(iii)

AlsotheinterceptsareNOTequal.

i.e(c1b1)(c2b2)c1c2b1b2......(iv)

Combining(iii)&(iv)wehavea1a2=b1b2c1c2

AnsOptionB.

171783_88876_ans_acf9f8a0a24844758a8e7b6458ee326d.png

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