If the sum of slopes of lines represented by $ax^2+8xy+5y^2=0$ is twice their product,then $a=$

The separate equations of the lines represented by the pair of lines equation $x^2 + xy - 12y^2 = 0$ are

If the lines $x^2-4xy+y^2=0$ make angles $\alpha$ and $\beta$ with the positive direction of the $X$-axis,then $\cot^2 \alpha + \cot^2 \beta = $

The joint equation of the pair of lines passing through the origin and making an angle of $\frac{\pi}{4}$ with the line $3x + y - 6 = 0$ is

If one of the lines represented by the equation $ax^2 + 2hxy + by^2 = 0$ is coincident with one of the lines represented by $a'x^2 + 2h'xy + b'y^2 = 0$,then:

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