KC Sinha Solution Class 12 Chapter 29 3D Geometry : Straight Lines (त्रिविमीय ज्यामिति : सरल रेखा ) Exercise 29.2 (Q1-Q21)

 Exercise 29.2

TYPE-I

Question 1

If the lines $\frac{x-1}{-3}=\frac{y-2}{2 k}=\frac{z-3}{2}$ and $\quad \frac{x-1}{3 k}=\frac{y-1}{1}=\frac{z-6}{-5}$ are perpendicular, Find the value of k.

Sol :

Question 2

Show that the line joining the origin to the point $(2,1,1)$ is perpendicular to the line determined by the points (3,5,-1) and (4,3,-1)

Sol :

Question 3

If the coordinates of the points A,B,C,D be (1,2,3),(4,5,7),(-4,3,-6) and (2,9,2) respectively, then find the angle between AB and CD.

Sol :

Question 4

Find the angle between the lines whose direction ratios are a, b, c and b-c, c-a, a-b

Sol :

Question 5

show that the three lines with direction cosines

$\frac{12}{13}, -\frac{3}{13},-\frac{4}{13}; \frac{4}{13}, \frac{12}{13}, \frac{3}{13} ; \frac{3}{13},-\frac{4}{13}, \frac{12}{13}$ are mutually perpendicular.

Sol :

Question 6

Show that the line through the points (4,7,8) and (2,3,4) is parallel to the line through the points (-1,-2,1) and (1,2,5)

Sol :

Question 7

Show that the line through the points (1,-1,2) and (3,4,-2) is perpendicular to the line through the points (0,3,2) and (3,5,6)

Sol :

Question 8

Show that the line $\frac{x-5}{7}=\frac{y+2}{-5}=\frac{z}{1}$ and $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ are perpendicular to each other.

Sol :

Question 9

Find the angle between the following pair of lines :

(i) $\vec{r}=3 \hat{i}+\hat{j}-2 \hat{k}+\lambda(\hat{i}-\hat{j}-2 \hat{k})$ and $\vec{r}=2 \hat{i}-\hat{j}-56 \hat{k}+\mu(3 \hat{i}-5 \hat{j}-4 \hat{k})$

(ii) $\vec{r}=2 \hat{i}-5 \hat{j}+\hat{k}+\lambda(3 \hat{i}+2 \hat{j}+6 \hat{k})$ and $\vec{r}=7 \hat{\imath}-6 \hat{k}+\mu(\hat{i}+2 \hat{j}+2 \hat{k})$

Sol :

Question 10

Find the angle between the following pair of lines :

(i) $\frac{x}{2}=\frac{y}{2}=\frac{z}{1}$ and $\frac{x-5}{4}=\frac{y-2}{1}=\frac{z-3}{8}$

(ii) $\frac{x-2}{2}=\frac{y-1}{5}=\frac{z+3}{-3}$ and $\frac{x+2}{-1}=\frac{y-4}{8}=\frac{z-5}{4}$

Sol :

Question 11

Find the value of p so that the lines $\frac{1-x}{3}=\frac{7 y-14}{2 p}=\frac{z-3}{2}$ and $\frac{7-7 x}{3 p}=\frac{y-5}{1}=\frac{6-z}{5}$ are at right angles.

Sol :

Question 12

If $l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}, n_{2}$ are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are $m_{1} n_{2}-m_{2} n_{1}, n_{1}-l_{2}-n_{2} l_{1}, l_{1} m_{2}-l_{2} m_{1} .$

Sol :

TYPE-II

Question 13

Find the equation of a line parallel to x-axis and passing through the origin.

Sol :

Question 14

Find the equation of the line which passes through the point (1,2,3) and is parallel to the vector $3 \hat{i}+2 \hat{j}-2 \hat{k}$.

Sol :

Question 15

The cartesian equation of a line is $\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}$. Write its vector equation.

Sol :

Question 16

Find the vector and cartesian equation of the line that passes through the origin and (5,-2,3).

Sol :

Question 17

Find the vector equation of the line passing through the points (-1,0,2) and (3,4,6).

Sol :

Question 18

The cartesian equation of a line is $\frac{x+3}{2}=\frac{y-5}{4}=\frac{z+6}{2}$

Find its vector equation.

Sol :

Question 19

Find the equation of the line in cartesian form that passes through the point with position vector $2 \hat{i}-\hat{j}+4 \hat{k}$ and is in the direction $\hat{i}+2 \hat{j}-\hat{k}$.

Sol :

TYPE-III : (Problems on shortest distance between two lines)

Question 20

Find the shortest distance between the following pair of lines :

(i)

$\vec{r}=\hat{i}+\hat{j}+\lambda(2 \hat{i}-\hat{j}+\hat{k})$ and

$\vec{r}=2 \hat{i}+\hat{j}-\hat{k}+\mu(3 \hat{i}-5 \hat{j}+2 \hat{k})$

(ii)

$\vec{r}=\hat{i}+2 \hat{j}+\hat{k}+\lambda(\hat{i}-\hat{j}+\hat{k})$ and

$\vec{r}=2 \hat{i}-\hat{j}-\hat{k}+\mu(2 \hat{i}+\hat{j}+2 \hat{k})$

(iii)

$\vec{r}=(1-t) \hat{i}+(t-2) \hat{j}+(3-2 t) \hat{k}$

$\vec{r}=(s+1) \hat{i}+(2 s-1) \hat{j}-(2 s+1) \hat{k}$

(iv)

$\vec{r}=\hat{i}+2 \hat{j}+3 \hat{k}+\lambda(\hat{i}-3 \hat{j}+2 \hat{k})$ and

$\vec{r}=4 \hat{i}+5 \hat{j}+6 \hat{k}+\mu(2 \hat{i}+3 \hat{j}+\hat{k})$

(v)

$\vec{r}=\hat{i}+2 \hat{j}-4 \hat{k}+\lambda(2 \hat{i}+3 \hat{j}+6 \hat{k})$ and

$\vec{r}=3 \hat{i}+3 \hat{j}-5 \hat{k}+\mu(2 \hat{i}+3 \hat{j}+6 \hat{k})$

Question 21

Find the shortest distance between the following pair of lines :

$\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}$ and $\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}$

Sol :