The joint equation of a pair of straight lines passing through the origin and having slopes $(1+\sqrt{2})$ and $\left(\frac{1}{1+\sqrt{2}}\right)$ is $......$

The area of the triangle formed by the lines $3x^2 - 4xy + y^2 = 0$ and $2x - y = 6$ is:

If $s$ and $p$ are respectively the sum and the product of the slopes of the lines $3x^2 - 2xy - 15y^2 = 0$,then $s:p$ is equal to

Let $L_1, L_2$ be the lines represented by the equation $4x^2-5xy+3y^2=0$. Let $L_3, L_4$ be two lines passing through the point $(4,3)$ such that $L_3$ and $L_4$ are perpendicular to $L_1$ and $L_2$ respectively. If the combined equation of $L_3$ and $L_4$ is $ax^2+2hxy+by^2+2gx+2fy+c=0$,then find the value of $af+bg+ch$.

If $4ab = 3h^2$,then the ratio of slopes of the lines represented by $ax^2 + 2hxy + by^2 = 0$ is

The combined equation of a pair of lines passing through the origin and inclined at $60^{\circ}$ and $30^{\circ}$ respectively with the $x$-axis is: