General Second Degree Equation in x and y

$∆OAB$ is formed by the lines $x^2-4xy+y^2=0$ and the line $AB.$ The equation of line  $AB$ is $2x+3y-1=0 .$  Find the equation of the median of the triangle drawn from the origin.
 

Equations of pairs of opposite sides of a parallelogram are $x^2-7x+6=0$ and $y^2-14y+40=0.$  Find the joint equation of its diagonals.
 

Find $p$ and $q$, if the equation $2x^2+8xy+p{y}^2+qx+2y-15=0$  represents a pair of parallel lines.

Find $p$ and $q$, if the equation $p{x}^2-8xy+3y^2+14x+2y+q=0$  represents a pair of perpendicular lines.

Find the value of $k$ if the following equations represent a pair of lines:

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

Find the value of $k$ if the following equations represent a pair of lines:

$kxy+10x+6y+4=0$

Find the separate equation of the lines represented by the following equations:

$10{\left(x+1\right)}^2+\left(x+1\right)\left(y-2\right)-3{\left(y-2\right)}^2=0$

Show that the following equations represent a pair of lines. Find the acute angle between them.

$2x^2+xy-y^2+x+4y-3=0$

Show that equation $2x^2-xy-3y^2-6x+19y-20=0$ represents a pair of lines.

Find the combined equation of the following pairs of lines:

$x+2y-1=0$ and $x-3y+2=0$

Find the co-ordinates of the point of intersection of the lines represented by $x^2-y^2-2x+1=0.$

Find the combined equation of the bisectors of the angles between the lines represented by $5x^2+6xy-y^2=0.$

The area of triangle formed by the lines $x^2+4xy+y^2=0$ and $x-y-4=0$ is _____

Find the value of $k$ if the following equations represent a pair of lines:

$3x^2+10xy+3y^2+16y+k=0$

Find the separate equation of the lines represented by the following equations:

${\left(x-2\right)}^2-3\left(x-2\right)\left(y+1\right)+2{\left(y+1\right)}^2=0$

Show that equation $x^2+2xy+2y^2+2x+2y+1=0$ does not represent a pair of lines.

Find the joint equation of the pair of lines:

Through the point $\left(2, -3\right)$ and parallel to lines represented by $x^2+xy-y^2=0$.

Find the joint equation of the pair of lines:

Through the point $\left(2, -1\right)$ and parallel to lines represented by $2x^2+3xy-9y^2=0$.

Find the condition that the equation  $a{y}^2+bxy+ex+dy=0$  may represent a pair of lines.

Show that the following equations represent a pair of lines. Find the acute angle between them.

$9x^2-6xy+y^2+18x-6y+8=0$