The area (in square units) of the quadrilateral formed by the two pairs of lines $l^2x^2 - m^2y^2 - n(lx + my) = 0$ and $l^2x^2 - m^2y^2 + n(lx - my) = 0$ is

The area (in sq units) of the quadrilateral formed by two pairs of lines $\lambda^2 x^2 - m^2 y^2 - n(\lambda x + my) = 0$ and $\lambda^2 x^2 - m^2 y^2 + n(\lambda x + my) = 0$ is:

The numbers $\alpha$ and $\beta$ are such that one of the lines of $2x^2 + \alpha xy + 3y^2 = 0$ coincides with one of the lines of $2x^2 + \beta xy - 3y^2 = 0$. If the two lines other than that common line are perpendicular,then $|\alpha + \beta|$ is equal to

If $(a, b)$ is the centroid of the triangle formed by the lines $4x^2 - 17xy + 4y^2 = 0$ and $x + y - 5 = 0$,and $c$ is the numerical value of the area of the triangle,then $a + b + c =$

If $P$ is the set of all real numbers $\alpha$ such that the product of the lengths of perpendiculars from $(\alpha, 1)$ to the pair of straight lines $3x^2+7xy+2y^2=0$ is $\frac{\sqrt{2}}{5}$,then the sum of the elements in $P$ is

If $2x^2 - 3xy + y^2 = 0$ represents two sides of a triangle and $x + y - 1 = 0$ is its third side,then the distance between the orthocenter and the circumcentre of that triangle is