The equation of the straight line passing through the point of intersection of $5x - 6y - 1 = 0$ and $3x + 2y + 5 = 0$ and perpendicular to the line $3x - 5y + 11 = 0$ is

For $\lambda, \mu \in R$,$(x-2y-1)+\lambda(3x+2y-11)=0$ and $(3x+4y-11)+\mu(-x+2y-3)=0$ represent two families of lines. If the equation of the line common to both the families is $ax+by-5=0$,then $2a+b=$

Given the family of lines,$a(2x + y + 4) + b(x - 2y - 3) = 0$. Among the lines of the family,the number of lines situated at a distance of $\sqrt{10}$ from the point $M(2, -3)$ is:

The equation of the line joining the point $(3, 5)$ to the point of intersection of the lines $4x + y - 1 = 0$ and $7x - 3y - 35 = 0$ is equidistant from the points $(0, 0)$ and $(8, 34)$.

If the perpendicular bisector of the line segment joining $A(\alpha, 3)$ and $B(2, -1)$ has $y$-intercept $1$,then $\alpha =$

$A$ square of side $a$ lies above the $x$-axis and has one vertex at the origin. The side passing through the origin makes an angle $\alpha, (0 < \alpha < \frac{\pi}{4})$ with the positive direction of the $x$-axis. The equation of its diagonal not passing through the origin is