The equation $x^2 - 7xy + 12y^2 = 0$ represents

If the sum of the slopes of the lines given by $x^2-4pxy+8y^2=0$ is three times their product,then $p=\ldots$

The equation of the pair of straight lines passing through the point $(2,3)$ and perpendicular to the pair of lines $3x^2-4xy+5y^2=0$ is $ax^2+2hxy+by^2+2gx+2fy+c=0$. Then $a+b+c+f+g+h=$

The value of $\lambda$, for which the equation $x^2 - y^2 - x - \lambda y - 2 = 0$ represents a pair of straight lines, is

The combined equation of the lines passing through the point $(3,4)$ and each making an angle $45^{\circ}$ with the line $x+y+1=0$ is

The locus of a point which moves such that its distance from the origin is thrice its distance from the line $y = 2x$ is