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Question

If the equation ax2+by2+cx+cy=0, represents a pair of straight lines, then


A

a(b+c)=0

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B

b(c+a)=0

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C

c(a+b)=0

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D

a+b+c=0

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Solution

The correct option is C

c(a+b)=0


Explanation for correct option

Step 1: Comparing with the general equation of pair of straight line

Given equation is ax2+by2+cx+cy=0

The general equation of the pair of straight line is a'x2+2h'xy+b'y2+2g'x+2f'y+c'=0

Now comparing with the general equation of the straight line we get

a'=a,h'=0,b'=b,g'=c2,f'=c2,c'=0

Step 2: Solve for the condition

The necessary and sufficient condition for general equation a'x2+2h'xy+b'y2+2g'x+2f'y+c'=0 represent a pair of straight line is

a'b'c'+2f'g'h'-a'f'2-b'g'2-c'h'2=0

h'2>a'b'

g'2>a'c'

From the first condition we have,

a'b'c'+2f'g'h'-a'f'2-b'g'2-c'h'2=00+0-ac22-bc22-0=0ac2+bc2=0c2a+b=0ca+b=0

Hence, Option (C) is correct i.e. c(a+b)=0


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