The area of the region enclosed between the pair of lines $xy = 0$ and the lines $xy + 5x - 4y - 20 = 0$ is .....

If the pair of lines represented by $3x^2 - 5xy + Py^2 = 0$ and $6x^2 - xy - 5y^2 = 0$ have one line in common,then the sum of all possible values of $P$ is

The area bounded by the angle bisectors of the lines ${x^2} - {y^2} + 2y = 1$ and the line $x + y = 3$ is:

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

Assertion $(A)$: The lines $2x^2 + 5xy + 2y^2 = 0$ and $x - 2y + 1 = 0$ form a right-angled triangle.
Reason $(R)$: The equation $ax^2 + 2hxy + by^2 = 0$ represents a pair of perpendicular lines if $a + b = 0$.
Choose the correct answer.

The two pairs of straight lines $12x^2+7xy-12y^2=0$ and $12x^2+7xy-12y^2-x+7y-1=0$ constitute a