Select and write the correct answer from the alternatives in each of the following :
If an equation hxy ` + gx + fy + c = 0` represents a pair of lines, then ……

Options

A. `fg = ch`
B. `gh = cf`
C. `fh = cg`
D. `hf = - cg`

Answer

A

Explanation

The given equation `hxy + g x + f y + c = 0` will represent pair of straight line ,
If
`|(0 h/2 (-g)/2),(h/2 0 (-f)/2),((-g)/2 (-f)/2 c)| = 0`
Expand along `R_(1)`
We get,
`0*[0-f^(2)/2]-h/2 [(hc)/2-(gf)/4]-g/2[(-gh)/4-0] = 0`
` 0 - h/2 [(hc)/2- (gf)/4] - g/2 [(-fh)/4] = 0`
` - h/2 [(hc)/2 - (gf)/4] = g/2 [(-fh)/4]`
` - (h^(2)c)/4 + (ghf)/8 = (-gfh)/8`
` - (h^(2)c)/4 = (-gfh)/8 - (gfh)/8`
` - (h^(2)c)/4 = (-gfh)/4`
hc = gf
or ` fg = ch `
Hence, the correct answer from the given alternative is (a) .