Show that a homogeneous equations of degree two in x and y , i.e., `ax^(2) + 2 hxy + by^(2) = 0` represents a pair of lines passing through the origin if `h^(2) - 2ab ge 0`.

Explanation

Let us consider a homogeneous equation of degree two in x and y .
i.e., ` ax^(2) + 2 hxy + by^(2) = 0` ….(1)
where, either a or b or h is non - zero .
We consider two cases :
Case 1 : If b = 0 , then equation becomes
` ax^(2) + 2 hxy = 0`
` x (ax + 2hy) = 0`
which is a joint equation of lines x = 0 and ax + 2hy = 0 thus , the lines pass through the origin.
Case 2 : If ` b ne 0`.
Multiplying both sides of quation (1) by b , we get
` abx^(2) + 2hbxy + b^(2) y^(2) = 0`
`b^(2)y^(2) + 2hbxy = - abx^(2)`
`b^(2)y^(2) + 2 hbxy + h^(2)x^(2) = - abx^(2) + h^(2) x^(2)`
` (by +hx)^(2) = (h^(2)-ab) x^(2)`
`(by + hx)^(2) = [(sqrt(h^(2)-ab))x]^(2)`
`(by+hx)^(2)-[(sqrt(h^(2)-ab))x]^(2) = 0`
`[(by+hx)+(sqrt(h^(2)-ab))x][(by+hx)]-(sqrt(h^(2)-ab))x)`
= 0
which is a joint equation of two lines
`(by+hx)+(sqrt(h^(2)-ab)) x = 0`
and ` (by+hx)-(sqrt(h^(2)-ab)) x = 0`
i.e., `(h+sqrt(h^(2)-ab)) x + by = 0 and (h - sqrt(h^(2)-ab)) x +by = 0 `
These lines passes through the origin when `h^(2) - ab ge 0` .
Hence, we can say that the equation `ax^(2) + 2hxy + by^(2) = 0` represents a pair of lines passing through the origin.