The equation $(x - 5)^2 + (x - 5)(y - 6) - 2(y - 6)^2 = 0$ represents

If $x^2-y^2+2hxy+2gx+2fy+c=0$ is the locus of a point which moves such that it is always equidistant from the lines $x+2y+7=0$ and $2x-y+8=0$,then the value of $g+c+h-f$ equals

The values of $h$ for which the equation $3x^2 + 2hxy - 3y^2 - 40x + 30y - 75 = 0$ represents a pair of straight lines are:

The distance between the parallel lines $9x^2 - 6xy + y^2 + 18x - 6y + 8 = 0$ is

If the sum of the slopes of the lines given by $4x^2 + 2\lambda xy - 7y^2 = 0$ is equal to the product of the slopes,then $\lambda$ is equal to

Assertion $(A)$: The difference of the slopes of the lines represented by $y^2 - 2xy \sec^2 \alpha + (3 + \tan^2 \alpha)(\tan^2 \alpha - 1) x^2 = 0$ is $4$.
Reason $(R)$: The difference of the slopes of the lines represented by $ax^2 + 2hxy + by^2 = 0$ is $\frac{2 \sqrt{h^2 - ab}}{|b|}$.