Find the equation of the line perpendicular to $5x - 2y + 7 = 0$ and passing through the intersection of the lines $y = x + 7$ and $x + 2y + 1 = 0$.

If $4a^2 + 9b^2 - c^2 + 12ab = 0$, then the family of straight lines $ax + by + c = 0$ is concurrent at

The distance between the points of concurrency of the two families of straight lines given by $x+(5 \lambda+1) y+1-3 \lambda=0$ and $(5 \mu+2) x-3 y+3+6 \mu=0$ is

For all values of $a$ and $b$,the line $(a+2b)x + (a-b)y + (a+5b) = 0$ passes through a fixed point. Find that point.

For the family of lines given by the equation $(2 + k)x + (1 + k)y = 5 + 7k$,which of the following statements is true for different values of $k$?

If the lines $ax + y + 1 = 0$,$x + by + 1 = 0$ and $x + y + c = 0$ (where $a, b$ and $c$ are distinct and different from $1$) are concurrent,then the value of $\frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c} = $