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:

If $16a^2 - 40ab + 25b^2 - c^2 = 0$,then the line $ax + by + c = 0$ passes through which points?

$L_1 \equiv 2x+y-3=0$ and $L_2 \equiv ax+by+c=0$ are two equal sides of an isosceles triangle. If $L_3 \equiv x+2y+1=0$ is the third side of this triangle and $(5,1)$ is a point on $L_2=0$,then $\frac{b^2}{|ac|}=$

$A$ 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

If the intercept of a straight line $L$ made between the straight lines $5x - y - 4 = 0$ and $3x + 4y - 4 = 0$ is bisected at the point $(1, 5)$,then the equation of $L$ is

If $k_1 > k_2$ are the two values of $k$ such that the lines $y - 3kx + 4 = 0$ and $(2k - 1)x - (8k - 1)y - 6 = 0$ are perpendicular,then the equation of the line passing through $(k_1, k_2)$ and having the slope $\left(\frac{k_2}{k_1}\right)$ is