Homogeneous Equation of Degree Two

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

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

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 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 separate equations of the lines represented by following equations:

$x^2+2xy\tan \alpha -y^2=0$

Find the combined equation of the following pairs of lines:

$2x+y=0$ and $3x-y=0$

If equation $a{x}^2-y^2+2y+c=1$ represents a pair of perpendicular lines then find $a$ and $c$

If the lines given by $a{x}^2+2hxy+b{y}^2=0$ form an equilateral triangle with the line $l x+m y=1$ then show that $(3a+b)(a+3b)=4h^2.$

Determine the nature of lines represented by

$x^2-2\sqrt{3}xy-y^2=0$

If distance between lines ${\left(x-2y\right)}^2+k\left(x-2y\right)=0$ is 3 units, then $k=$ _____

The combined equation of the co-ordinate axes is _____

Find the value of $k$ if lines represented by $k{x}^2+4xy-4y^2=0$ are perpendicular to each other.

Show that lines represented by $3x^2-4xy-3y^2=0$ are perpendicular to each other.

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

Determine the nature of lines represented by

$x^2+7xy+2y^2=0$

Determine the nature of lines represented by

$x^2-y^2=0$

Determine the nature of lines represented by

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

Determine the nature of lines represented by

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

Find the joint equation of the lines:

Passing through the origin and perpendicular to the lines $x+2y=19$ and $3x+y=18 .$