If the equation of the base of an equilateral triangle is $2x - y = 1$ and the vertex is $(-1, 2)$,then the length of the side of the triangle is

The equation of the straight line perpendicular to $5x - 2y = 7$ and passing through the point of intersection of the lines $2x + 3y = 1$ and $3x + 4y = 6$ is

Let $A(-1, 1)$,$B(3, 4)$,and $C(2, 0)$ be three given points. $A$ line $y = mx$,$m > 0$,intersects lines $AC$ and $BC$ at points $P$ and $Q$ respectively. Let $A_1$ and $A_2$ be the areas of $\Delta ABC$ and $\Delta PQC$ respectively,such that $A_1 = 3A_2$. Then the value of $m$ is equal to:

The lines $x + y = |a|$ and $ax - y = 1$ intersect each other in the first quadrant. Then,in which interval do all possible values of $a$ lie?

Let the area of the triangle formed by a straight line $L : x + by + c = 0$ with the coordinate axes be $48$ square units. If the perpendicular drawn from the origin to the line $L$ makes an angle of $45^{\circ}$ with the positive $x$-axis,then the value of $b^2 + c^2$ is:

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