Question

Show that x^2+2xy-y^2=0 represents pair of line

Answer 1

Pair of Straight Lines

Rule. The general equation of second degree is

$\quad ax^{2}+2hxy+by^{2}+2gx+2fy+c=0$.

It will represent a pair of straight lines, if

$\quad abc+2fgh-af^{2}-bg^{2}-ch^{2}=0$ .....(1)

Proof.

The given equation in second degree is

$\quad x^{2}+2xy-y^{2}=0$.

Comparing it with the general equation of second degree is

$\quad a=1,\:h=1,\:b=-1,\:g=f=c=0$.

Putting these values in left hand side of (1) no. condition, we get

$\quad (1)(1)(0)+2(0)(0)(1)-(1)(0^{2})-(-1)(0^{2})-(0)(1^{2})$

$=0+0-0-0-0$

$=\bold{0}$

= right hand side of (1)

Hence the necessary condition for pair of straight lines is satisfied.

Therefore the equation $x^{2}+2xy-y^{2}=0$ represents a pair of straight lines.

This completes the proof.