The product of the perpendicular distances drawn from the origin to the pair of straight lines $6x^2 - 5xy - 6y^2 + x + 5y - 1 = 0$ is

If one of the lines of the pair $ax^{2}+2hxy+by^{2}=0$ bisects the angle between the positive direction of the axes,then $a, b,$ and $h$ satisfy the relation:

The joint equation of lines passing through the origin and having slopes $(1+\sqrt{2})$ and $\frac{-1}{1+\sqrt{2}}$ is ...

Let $PQR$ be a right-angled isosceles triangle,with the right angle at $Q(2, 1)$. If the equation of the line $PR$ is $2x + y = 3$,then the combined equation representing the pair of lines $PQ$ and $QR$ is:

The equation of the line common to the pair of lines $(p^2-q^2) x^2+(q^2-r^2) xy+(r^2-p^2) y^2=0$ and $(l-m) x^2+(m-n) xy+(n-l) y^2=0$ is

The locus of a point which moves such that its distance from the $x$-axis is twice its distance from the line $x-y=0$ is