If $u = a_1x + b_1y + c_1 = 0$,$v = a_2x + b_2y + c_2 = 0$ and $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$,then the curve $u + kv = 0$ is

In the figure,$AHKF$,$FKDE$ and $HBCK$ are unit squares. $AD$ and $BF$ intersect at $X$. Then,the ratio of the areas of triangles $AXF$ and $ABF$ is

Find the equation of the line which intersects the $y$-axis at a distance of $2$ units above the origin and makes an angle of $30^{\circ}$ with the positive direction of the $x$-axis.

The equations of the lines which make intercepts on the axes whose sum is $8$ and product is $15$ are

$\beta$ is the angle made by the perpendicular drawn from the origin to the line $L \equiv x+y-2=0$ with the positive $X$-axis in the anticlockwise direction. If '$a$' is the $X$-intercept of the line $L=0$ and $p$ is the perpendicular distance from the origin to the line $L=0$,then $a \tan \beta + p^2 =$

Find the equation of the line passing through the point $(-2, 3)$ with a slope of $-4$.