Direction Cosines of a Line Passing through Two Points

Q.

Let a plane $P$contain two lines $\begin{array}{l}\overrightarrow{r}=\hat{i}+\lambda (\hat{i}+\hat{j})\end{array}$, $\lambda \in R$ and $\begin{array}{l}\overrightarrow{r}=-\hat{j}+\mu (\hat{j}-\hat{k})\end{array}$, $\mu \in R$. If $Q(\alpha ,\beta ,\gamma )$ is the foot of the perpendicular drawn from the point $M(1,0,1)$ to $P$, then $3(\alpha +\beta +\gamma )$ equals:

Q.

The straight lines joining the origin to the points of intersection of the line $2x+y=1$ and curve $3x^2+4xy-4x+1=0$include an angle:

1. $\frac{\mathrm{\pi }}{2}$

2. $\frac{\mathrm{\pi }}{3}$

3. $\frac{\mathrm{\pi }}{4}$

4. $\frac{\mathrm{\pi }}{6}$

Q. The distance of the point A(–2, 3, 1) from the line PQ through P(–3, 5, 2) which make equal angles with the axes is

1. $\frac{2}{\sqrt{3}}$
2. $\sqrt{\frac{14}{3}}$
3. $\frac{16}{\sqrt{3}}$
4. $\frac{5}{\sqrt{3}}$

Q. Find the points on the line 3x-4y-1=0 which are at a distance of 5 units from point (3, 2)?

Q.

If $g\left(x\right)=\left\{\begin{array}{l}2e\mathrm{if}\mathrm{x}\le 1\\ \log \left(x-1\right),\mathrm{if}x>1\end{array}\right.$, then the equation of the normal to $y=g\left(x\right)$ at the point $\left(3,\log 2\right)$ is

1. $y-2x=6+\log 2$

2. $y+2x=6+\log 2$

3. $y+2x=6-\log 2$

4. $y+2x=-6+\log 2$

5. $y-2x=-6+\log 2$

Q.

A line with directiion ratios 2, 7, -5 is intercepted between the lines
$\frac{x-5}{3}=\frac{y-7}{-1}=\frac{z+2}{1}and\frac{x+3}{-3}=\frac{y-3}{2}=\frac{z-6}{4}$. Find the length intercepted between the given lines.___

1. $\sqrt{78}$

2. $\sqrt{85}$
3. 10
4. 9

Q. The distance of the point $A(-2,3,1)$ from the line $PQ$ through $P(-3,5,2)$ which make equal angles with axes is

1. $\frac{2}{\sqrt{3}}$
2. $\sqrt{\frac{14}{3}}$
3. $\frac{16}{\sqrt{3}}$
4. $\frac{5}{\sqrt{3}}$

Q. The limiting position of the point of intersection of the straight line 3x+5y=1 and (2+c)x+5c^2y=1 as c tends to one is (1) (1/2, -1/10) (2) (3/8, -1/40) (3) (2/5, 1/25) (4) (2/5, -1/25)

Q. 3. Let P be the plane passing through the point (2, 1, -1) and perpendicular to the line of intersection of the planes 2x+y-z=3 and x+2y+z=2. Then the distance from the point (\sqrt{}3, 2, 2) to the plane P is

Q.

Find the equation of the lines through the point of intersection of the lines x-3y+1 = 0 and 2x+5y-9 = 0 and whose distance from the origin is $\sqrt{5}$.

1. x-2y+5 = 0

2. 2x+y+5 = 0

3. 

4. 2x+y-5 = 0

Q.

The distance of the point (1, -5, 9) from the plane x - y + z = 5 measured along the line x = y = z is

1. $3\sqrt{10}$

2. $10\sqrt{3}$

3. $\frac{10}{\sqrt{3}}$

4. $\frac{20}{3}$

Q.

Minimise and Maximise Z = 5x + 10y

subject to.

Q. Find the direction cosines of the line passing through the points
P (2, 3, 5) and Q (–1, 2, 4).

1. (-2, -1, -1)
2. $\left(\frac{3}{\sqrt{11}},\frac{1}{\sqrt{11}},\frac{1}{\sqrt{11}}\right)$
3. $\left(\frac{-3}{\sqrt{11}},\frac{-1}{\sqrt{11}},\frac{-1}{\sqrt{11}}\right)$
4. $\left(\frac{-3}{\sqrt{11}},\frac{1}{\sqrt{11}},\frac{1}{\sqrt{11}}\right)$

Q. 22. The distance of the point (3, 8, 2) from the line x-1/2 = y-3/4 = z-2/3 measured parallel to the plane 3x+2y-2z+15=0 is?

Q. The distance of the point $(1,1)$ from the line $2x-3y-4=0$ measured in the direction of the line $x+y=1$ is

1. $\sqrt{2}$ units
2. $5\sqrt{2}$ units
3. $\frac{1}{\sqrt{2}}$ units
4. $2\sqrt{2}$ units

Q.

$\begin{array}{cccc}& \text{Column I}& & \text{Column II}\\ \text{(a)}& arg\left(\frac{z+1}{z-1}\right)=\frac{\pi }{4}& \text{(p)}& \text{Parabola}\\ \text{(b)}& z=\frac{3i-1}{2+it}\left(tϵR\right)& \text{(q)}& \text{Part of a circle}\\ \text{(c)}& argz=\frac{\pi }{4}& \text{(r)}& \text{Full circle}\\ \text{(d)}& z=t+i{t}^2\left(tϵR\right)& \text{(s)}& \text{Line}\end{array}$

Which of the following is correct?

1. A-Q, B-R, C-S, D-Q

2. A-S, B-R, C-P, D-Q

3. A-Q, B-R, C-S, D-P

4. A-R, B-S, C-P, D-Q

Q. The direction cosines of the line drawn from P(-5, 3, 1) to Q(1, 5, -2) is

1. $(6,2,-3)$
2. $(2,-4,1)$
3. $(-4,8,-1)$
4. $(\frac{6}{7},\frac{2}{7},-\frac{3}{7})$

Q.

The direction cosines of the line whose direction ratios are 6, - 6, 3 are:

1. 

2. 

3. 

4. 

Q. The perpendicular distance of the point P (1, 2, 3) from the line $\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}$ is
(a) 7
(b) 5
(c) 0
(d) none of these

Q. Suppose the direction cosines of two lines are given by $al+bm+cn=0$ and $fmn+gln+hlm=0$ where $f,g,h,a,b,c$ are arbitrary constants and $l,m,n$ are direction cosines of the line. On the basis of the above information and for $f=g=h=1$ both lines satisfy the relation:

1. $a{\left(\frac{l}{m}\right)}^2+(a+b-c)\left(\frac{l}{m}\right)+b=0$
2. $b{\left(\frac{m}{n}\right)}^2+(b+c-a)\left(\frac{m}{n}\right)+c=0$
3. $c{\left(\frac{n}{l}\right)}^2+(c+a-b)\frac{n}{l}+a=0$
4. All of the above.

Q. Find the direction cosines of the line passing through two points (−2, 4, −5) and (1, 2, 3).

Q. Find the distance of the point with position vector $-\hat{i}-5\hat{j}-10\hat{k}$ from the point of intersection of the line $\overrightarrow{r}=\left(2\hat{i}-\hat{j}+2\hat{k}\right)+\lambda \left(3\hat{i}+4\hat{j}+12\hat{k}\right)$ with the plane $\overrightarrow{r}·\left(\hat{i}-\hat{j}+\hat{k}\right)=5.$

Q. The distance of the point (−1, −5, −10) from the point of intersection of the line $\overrightarrow{r}=2\hat{i}-\hat{j}+2\hat{k}+\lambda \left(3\hat{i}+4\hat{j}+12\hat{k}\right)$ and the plane $\overrightarrow{r}·\left(\hat{i}-\hat{j}+\hat{k}\right)=5$ is
(a) 9
(b) 13
(c) 17
(d) None of these

Q. A line passes through the points $(6,-7,-1)$ and $(2,-3,1)$. If the angle $\alpha$ which the line makes with the positive direction of x-axis is acute, the direction cosines of the line are

1. $\frac{2}{3},\frac{-2}{3},\frac{-1}{3}$
2. $\frac{2}{3},\frac{2}{3},\frac{-1}{3}$
3. $\frac{2}{3},\frac{-2}{3},\frac{1}{3}$
4. $\frac{2}{3},\frac{2}{3},\frac{1}{3}$

Q.

The distance of the point (1, -5, 9) from the plane x - y + z = 5 measured along the line x = y = z is

1. $3\sqrt{10}$

2. $10\sqrt{3}$

3. $\frac{10}{\sqrt{3}}$

4. $\frac{20}{3}$

Q. The direction cosines of the line drawn from P(–5, 3, 1) to Q(1, 5, –2) is

1. $(-4,8,-1)$
2. $(\frac{6}{7},\frac{2}{7},-\frac{3}{7})$
3. $(6,2,-3)$
4. $(2,-4,1)$

Q. Find the image of the point with position vector $3\hat{i}+\hat{j}+2\hat{k}$ in the plane $\overrightarrow{r}·\left(2\hat{i}-\hat{j}+\hat{k}\right)=4.$ Also, find the position vectors of the foot of the perpendicular and the equation of the perpendicular line through $3\hat{i}+\hat{j}+2\hat{k}.$

Q. The length of the projection of the line segement joining the points $(5,-1,4)$ and $(4,-1,3)$ on the plane, $x+y+z=7$ is

1. $\sqrt{\frac{2}{3}}$
2. $\frac{2}{\sqrt{3}}$
3. $\frac{2}{3}$
4. $\frac{1}{3}$

Q. If the line joining the points $(k,2,3),(1,1,2)$ is parallel to the line joining the points $(5,4,-1),(3,2,-3)$ then the value of $k$ is:

Q. Find the distance of the point (2, 12, 5) from the point of intersection of the line $\overrightarrow{r}=2\hat{i}-4\hat{j}+2\hat{k}+\lambda \left(3\hat{i}+4\hat{j}+2\hat{k}\right)$ and $\overrightarrow{r}.\left(\hat{i}-2\hat{j}+\hat{k}\right)=0$. [CBSE 2014]