Consider the following lines in the $X Y$-plane:

$$ L_1: 5 x-2 y=1 $$

$L_2$ : The line passing through $(0,1)$ and $(100,101)$,

$L_3$ : The line passing through $(1,11)$ and parallel to the vector $-\hat{\imath}+2 \hat{\jmath}$.

Let $A=\left(L_1 \cap L_2\right) \cup\left(L_2 \cap L_3\right) \cup\left(L_3 \cap L_1\right)$. What is the total number of elements of $A$ ?

Let $A$ be the set of points in the $X Y$-plane which are equidistant from $P(-1,0)$ and $Q(1,0)$. Let $B$ be the set of points in the $X Y$-plane which are equidistant from $A$ and $Q$. If $(5, y)$ is a point in $B$, then what is the value of $y^2$ ?

Consider the lines $L_1$ and $L_2$ given below:

$$ \begin{gathered} L 1: x=2+\lambda, y=3+2 \lambda, z=4+3 \lambda \\ L 2: x=4+\lambda, y=4, z=4+\lambda \end{gathered} $$

If $(2,3,4)$ is the point of $L_1$ that is closest to $L_2$, then which point of $L_2$ is closest to $L_1$ ?

Let $a_1, a_2, a_3, \ldots$ be a sequence of real numbers. Let $s_n=a_1+a_2+\cdots+a_n$. If $2 s_n=$ $n\left(c+a_n\right)$ for some real number. for some real number $c$ and for all $n=1,2,3, \ldots$, then which one of the following statements is Correct?