Combined Equation of Two line

Show that the two straight lines
${\mathrm{x}}^2\left({\tan }^2\mathrm{\theta }+{\cos }^2\mathrm{\theta }\right)-2\mathrm{x}\mathrm{y}\tan \mathrm{\theta }+{\mathrm{y}}^2{\sin }^2\mathrm{\theta }=0$
Makes with the axis of $\mathrm{x}$ angles such that the difference of their tangents is $2.$

The equation of the line common to the pair of lines $m^2{x}^2-\left(m^2+1\right)xy+y^2=0$ and $m{x}^2-\left(m+1\right)xy+y^2=0$ is:

Equation of all lines passing through the intersection of lines $2x+3y=5$ and $x+y=2$ is given by $(2x+3y-5)+\lambda (x+y-2)=0$, of all these lines one line is farthest from point $(7,4)$, the value of $\lambda$ is :

If a line $L$ is common to the pairs of lines $6x^2-xy-12y^2=0$ and $15x^2+14xy-8y^2=0$, then the combined equation the other two lines is

Let $a$ and $b$ be non-zero real numbers. Then, the equation $\left(a{x}^2+b{y}^2+c\right)\left(x^2-5xy+6y^2\right)=0$ represents 

Find the joint equations of lines passing through origin and having inclinations $\frac{\mathrm{\pi }}{3} \mathrm{and} \frac{5\mathrm{\pi }}{3}$

If a pair of lines drawn through the origin forms an isosceles right-angled triangle with the line $2x+3y=6$, then those lines are

If the product of the perpendiculars from origin to the pairs of lines $xy+x+y+1=0,$ $x^2-y^2+2x+1=0$ and $2x^2+3xy-2y^2+2x+1=0$ respectively are $p_1, p_2, p_3,$ then

The distance between pair of parallel lines represented by $x^2+2xy+y^2-8ax-8ay-9a^2=0$ is _____ units

If slopes of lines represented by $K{x}^2+5xy+y^2=0$ differ by $\text{1}$ then $K=$

If lines represented by equation $p{x}^2-q{y}^2=0$ are distinct then

Find the product of the lengths of the perpendiculars drawn from $(2,1)$ upon the lines $12x^2+25xy+12y^2+10x+11y+2=0$.

The equation $6x^2-\alpha xy-3y^2-24x+3y+\beta =0$ represents a pair of straight lines intersecting on $x$-axis. Find the value of $20\alpha -\beta$.

The joint equation of the lines passing through $(1, 2)$ and parallel to the co-ordinate axes is

The joint equation of the lines passing through the origin having inclinations ${60}^{\circ }$ and ${120}^{\circ }$ with $x$-axis is

The joint equation of pair of lines through $\left(2,-1\right)$ and parallel to $2x^2+3xy-9y^2=0$ is

Find the joint equation of pair of lines through the origin each of which makes an angle of ${60}^{\circ }$ with the $Y$-axis.
 

$∆OAB$ 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 the separate equations of the lines represented by the following equation :

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