If the points $(1, 2)$ and $(3, 4)$ lie on opposite sides of the line $3x - 5y + a = 0$,then:

The length of the perpendicular from the point $(a \cos \alpha, a \sin \alpha)$ to the line $y = x \tan \alpha + c, c > 0$ is .....

The coordinates of a point on the line $x+y+1=0,$ which is at a distance of $\frac{1}{5}$ unit from the line $3x+4y+2=0,$ are

If $p$ and $q$ are the perpendicular distances from the origin to the straight lines $x \sec \theta - y \operatorname{cosec} \theta = a$ and $x \cos \theta + y \sin \theta = a \cos 2 \theta$,then

$O(0,0), B(-3,-1), C(-1,-3)$ are vertices of a triangle $OBC$. $D$ is a point on $OC$ and $E$ is a point on $OB$. If the equation of $DE$ is $2x + 2y + \sqrt{2} = 0$,then the ratio in which the line $DE$ divides the altitude of the triangle $OBC$ from $O$ to $BC$ is:

Let the lines be $L_1: 2x + 3y - 7 = 0$ and $L_2: 2x + 3y - 12 = 0$. For the point $A(3, -5)$,which of the following is true?