#### Question

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

(a) fg = ch (b) gh = cf

(c) Jh = cg (d) hf= - eg

#### Solution

**(a)**

Consider the general equation in second degree,

a'x^{ 2}+2h'xy + b'y^{2} + 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]`