If the pair of lines [ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0] intersect on the y-axis then…

Question

If the pair of lines

$$a{x^2} + 2hxy + b{y^2} + 2gx + 2fy + c = 0$$

intersect on the $y$-axis then :

Accepted Answer

Put $x=0$ in the given equation

$$ \Rightarrow b{y^2} + 2fy + c = 0.$$

For unique point of intersection ${f^2} - bc = 0$

$$ \Rightarrow a{f^2} - abc = 0.$$

Since $$abc + 2fgh - a{f^2} - b{g^2} - c{h^2} = 0$$

$$ \Rightarrow 2fgh - b{g^2} - c{h^2} = 0$$

Solutions · 1