Find the joint equation of a pair of lines through the origin, which are perpendicular to the lines represented by:

$5x^2+2xy-3y^2=0$

Find the combined equation of a pair of lines passing through the origin and perpendicular to the lines represented by following equations:

$xy+y^2=0$

Find the combined equation of a pair of lines passing through the origin and perpendicular to the lines represented by following equations:

$3x^2-4xy=0$

Find $k$ if the sum of the slopes of the lines represented by $x^2+kxy-3y^2=0$ is twice their product.

Find $k$ if slopes of lines represent by $3x^2+kxy-y^2=0$ differ by $4$.

Find $k$ if slope of one of the lines given by $k{x}^2+4xy-y^2=0$ exceeds the slope of the other by $8$ .

Find the condition that the line $4x+5y=0$ coincides with one of the lines given by $a{x}^2+2hxy+b{y}^2=0$.

Find the condition that the line $3x+y=0$ may be perpendicular to one of the lines given by $a{x}^2+2hxy+b{y}^2=0.$