Question

If an equation hxy + gx + fy + c = 0 represents a pair of lines, then __________

Options

• fg = ch         

• gh = cf

• Jh = cg     

• hf= - eg

Answer 1

(a)

Consider the general equation in second degree,
a'x ²+2h'xy + b'y² + 2g'x + 2f'y + c' = 0
The above equation will represent a pair of straight lines if,
a'f'² + b'g'² + c'h'² = 2f'g'h' + a'b'c'....(1)
Here, the given equation is, hxy + gx + fy + c = 0
Thus, comparing the coefficients, we have,
a' = 0, b' = 0, c' = 0, h' =h/2 , g' =g/2 , f' =f/2 , c' = c/2
Substituting the above values in the condition (1),
we have,

`(0)(f/2)^2+(0)(g/2)^2+(c)(h/2)^2=2xxf/2xxg/2xxh/2+(0)xx(0)xx(0)`

`(c)(h/2)^2=2xxf/2xxg/2xxh/2`

`(ch^2)/4=(fgh)/4`

`ch^2=fgh`

`ch=fg [because h!=0]`