If the equation of the straight line passing through the point of intersection of $x+2 y-19=0, x-2 y-3=0$ and which is at a perpendicular distance of 5 units from the point $(-2,4)$ is $5 x+b y+c=0$, then a possible value of $5+b+c$ is

Suppose $O(0,0)$ is the origin and the line $L=x+y-\lambda=0$ meets the curve $x^2+y^2-2 x-4 y+2=0$ at $A$ and $B$. If $\angle A O B=90^{\circ}$, then the distance between such lines $L=0$ is

Let $P$ be the point of intersection of the lines $L_1 \equiv x-y-7=0$ and $L_2 \equiv x+y-5=0 . A\left(x_1, y_1\right)$ and $B\left(x_2, y_2\right)$ are points on the lines $L_1=0$ and $L_2=0$ respectively such that $P A=3 \sqrt{2}$, $P B=\sqrt{2}, x_1, y_1 \geq 0, x_2, y_2 \geq 0$, then the angle made by the line segment $A B$ at the origin is

Let $A(2,1)$ be a point and equation of the straight line $L$ be $x-y=0$. Let $a$ and $b$ respectively represent the distances from a variable point $P(\alpha, \beta)$ to $A$ and to the line $L$. If $C$ is distance of the point $A$ from origin such that $a=b c$, then locus of $P$ is