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Question

What is the condition for the roots of ax2+bx+c=0 to be in the ratio of p : q ?

A
a2c2(p+q)=p2q2b
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B
ac(p+q)2=pqb2
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C
pq(a+c)2=acb2
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D
p2q2(a+c)=a2c2b
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Solution

The correct option is B ac(p+q)2=pqb2
Let the roots of ax2+bx+c=0 be α and β
α+β=ba
αβ=ca
Let αβ=pq
b+b24ac2abb24ac2a=pq
b+b24acbb24ac=pq
bq+qb24ac=pbpb24ac
(p+q)b24ac=bqpb
(p+q)2(b24ac)=b2q2+p2b22pqb2
p2b24acp2+q2b24acq28acpq+2pqb2=b2q2+p2b22pqb2
4ac(p+q)2=4pqb2
ac(p+q)2=pqb2

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