Find the area of the region bounded by the curve $4 x^2$ $+y^2=36$ using integration.

(a) Find the co-ordinates of the foot of the perpendicular drawn from the point $(2,3,-8)$ to the line $\frac{4-x}{2}=\frac{y}{6}=\frac{1-z}{3}$.

OR

(b). Find the shortest distance between the lines $\mathrm{L}_1 \& \mathrm{~L}_2$ given below: $\mathrm{L}_1$ : The line passing through $(2,-1,1)$ and parallel to $\frac{x}{1}=\frac{y}{1}=\frac{z}{3} \mathrm{~L}_2: \vec{r}=\hat{i}+(2 \mu+1) \hat{j}-(\mu+2) \hat{k}$.

$$\begin{aligned} &\text { Solve the following L. P. P. graphically: }\\ &\begin{array}{rlrl} \text { Maximise } & Z =60 x+40 y \\ \text { Subject to } & x+2 y \leq 12 \\ & 2 x+y \leq 12 \\ & 4 x+5 y \geq 20 \\ & x, y \geq 0 \end{array} \end{aligned}$$

(a) Students of a school are taken to a railway museum to learn about railways heritage and its history.

An exhibit in the museum depicted many rail lines on the track near the railway station. Let $L$ be the set of all rail lines on the railway track and $R$ be the relation on $L$ defined by

$R=\left\{l_1, l_2\right): l_1$ is parallel to $\left.l_2\right\}$

On the basis of the above information, answer the following questions:

(i) Find whether the relation R is symmetric or not.

(ii) Find whether the relation R is transitive or not.

(iii) If one of the rail lines on the railway track is represented by the equation $y=3 x+2$, then find the set of rail lines in R related to it.

OR

(b) Let $S$ be the relation defined by $S=\left\{\left(l_1, l_2\right): l_1\right.$ is perpendicular to $l_2$ \} check whether the relation $S$ is symmetric and transitive.