The value of $k$ so that the equation $2x^2 + 5xy + 3y^2 + 6x + 7y + k = 0$ represents a pair of straight lines is:

If the equation of the pair of straight lines passing through the point $(1, 1)$ and perpendicular to the pair of lines $3x^2 + 11xy - 4y^2 = 0$ is $ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$ (where the constant term is $12$),then find the value of $2(a - h + b - g + f - 12)$.

The combined equation of a possible pair of adjacent sides of a square with area $16 \text{ square units}$ whose centre is the point of intersection of the lines $x+2y-3=0$ and $2x-y-1=0$ is

If the equation $ax^{2}+2hxy+by^{2}+2gx+2fy=0$ has one line as the bisector of the angle between the coordinate axes,then

If $m_1$ and $m_2$ $(m_1 > m_2)$ are the slopes of the lines represented by $5x^2 - 8xy + 3y^2 = 0$,then $m_1 : m_2$ equals

If $2x^2 + 3xy - 2y^2 - 5x + 2fy - 3 = 0$ represents a pair of straight lines,then one of the possible values of $f$ is: