Question

If a system of three linear equations `x+4a y+a=0,x+3b y+b=0,a n dx+2c y+c=0` is consistent, then prove that `a ,b , c` are in H.P.

Answer 1

Since system of equation is consistent we have
`Delta = |{:(1,,4a,,a),(1,,3b,,b),(1,,2c,,c):}|=0`
`" or " 1(3bc -2bc ) -4a(c-b)+a(2c-3b)=0`
`" or " bc-4ac+4ab+2ac -3ab=0`
" or " bc-2ac+ =2ac`
" or " bc+ ab=2ac`
`" or "(1)/(a)+(1)/(c)=(2)/(b).`
Thus a,b,c are in H.P.