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Question

If the equation ax2+2hxy+by2+2gx+2fy+c=0 represent a pair of straight lines, then prove that the equation to the third pair of straight lines passing through the points where these meet the axis is ax22hxy+by2+2gx+2fy+c+4fgcxy=0

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Solution

ax2+2hxy+by2+2gx+2fy+c=0

using the concept of family of lines , the other two lines will be

ax2+2hxy+by2+2gx+2fy+c+λxy=0.......(i)

For this to represent straight lines

Δ=0abc+2fghaf2bg2ch2=0ab+2fg2h+λ2af2bg2c(2h+λ2)2=0(abc+2fghaf2bg2ch2)+λfgcλ24chλ=00+λfgcλ24chλ=0cλ24λfg+chλ=0λ(cλ4+fgch)=0λ=0,4(fgch)cλ=4fgc4h

Substituting in (i)

ax2+2hxy+by2+2gx+2fy+c+(4fgc4h)xy=0ax22hxy+by2+2gx+2fy+c+4fgcxy=0

Hence proved.


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