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Question

A system of linear equations is given as follows :
a1x+b1y+c1=0
a2x+b2y+c2=0
Condition for two lines to have a unique solution is,

A
a1a2=b1b2
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B
a1a2b1b2
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C
a1a2=b1b2c1c2
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D
a1a2=b1b2=c1c2
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Solution

The correct option is C a1a2b1b2
Twoequationswillhaveauniquesolutionif,aftersolvingthem,thesolutionisvalidi.ethelinesactuallyintersectatapoint.

Lettherebeequationsa1x+b1y+c1=0..........(i)and
a2x+b2y+c2=0.........(ii).

Multiplying(i)bya2&(ii)bya1andeliminatingxbysubtractionweget,

y=a1c2a2c1a2b1a1b2.

AgainMultiplying(i)byb2&(ii)byb1andeliminatingybysubtractionweget

x=b2c1b1c2a2b1a1b2.

Itisclearthatthesolution(x,y)=(b1c2b2c1a2b1a1b2,b2c1b1c2a2b1a1b2)willbevalidifa2b1a1b20a1a2b1b2.

AnsOptionB.

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