Foot of Perpendicular of a Point on a Line

Q. Find the co-ordinates of the foot of perpendicular drawn from the point $A(-1,8,4)$ to the line joining the points $B(0,–1,3)$ and $C(2,–3,–1)$.

Q.

Find the coordinates of a point on y-axis which are at a distance offrom the point P (3, –2, 5).

Q. The foot of the perpendicular from (0, 2, 3) to the line $\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}$ is :

1. (3, 2, –1)
2. (–2, 3, 4)
3. (2, –1, 3)
4. (2, 3, –1)

Q.

If $(-2,6)$ is the image of the point $(4,2)$ with respect to line $L=0$, then $L=$

1. $3x-2y+5$

2. $3x-2y+10$

3. $2x+3y-5$

4. $6x-4y-7$

Q.

If A, B, C are points (1, 0, 4) (0, -1, 5) and (2, -3, 1) respectively, then the coordinates of the foot of the perpendicular drawn from A to the line BC are

1. None of these

2. $(\frac{1}{2},-\frac{1}{2},\frac{9}{2})$

3. $(1,-2,3)$

4. $(\frac{3}{2},-\frac{3}{2},\frac{5}{2})$

Q.

The point P is the intersection of the straight line joining the points Q (2, 3, 5) and R (1, -1, 4) with the plane 5x-4y -z = 1. If S is the foot of the perpendicular drawn from the point T (2, 1 , 4) to QR, then the length of the line segment PS is

1. 2

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

3. $\sqrt{2}$
   

4. $2\sqrt{2}$

Q. The distance of the point $P(-2,3,-4)$ from the line $\frac{x+2}{3}=\frac{2y+3}{4}=\frac{3z+4}{5}$ measured parallel to the plane $4x+12y-3z+1=0$ is $d$, then the value of $(2d-8)$ is

Q. The foot of the perpendicular from (0, 2, 3) to the line $\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}$ is:

1. (2, 3, –1)
2. (3, 2, –1)
3. (–2, 3, 4)
4. (2, –1, 3)

Q. P, Q, R are the points of intersection of a line / with the sides BC, CA, AB of a$\Delta$ ABC respectively, then $\frac{BP}{PC}.\frac{CQ}{QA}.\frac{AR}{RB}$is

1. -1
2. 1
3. 2
4. -2

Q.

The coordinates of the foot of the perpendicular from a point P(6, 7, 8)

on x-axis are

1. (6, 0, 0)

2. (0, 7, 0)

3. (0, 0, 8)

4. (0, 7, 8)

Q.

Find the co-ordinates of the foot of perpendicular drawn from the point A (1, 8, 4) to the line joining the points B (0, -1, 3) and C (2, -3, -1)___

1. $(1,-2,1)$
2. $(2,-3,-1)$

3. $(\frac{-5}{3},\frac{2}{3},\frac{19}{3})$
   

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

Q.

Find the coordinates of a point on y-axis which are at a distance offrom the point P (3, –2, 5).

Q. The foot of the perpendicular from P(1, 0, 2) to the line $\frac{x+1}{3}=\frac{y-2}{-2}=\frac{z+1}{-1}$ is

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

Q.

Let the foot of the perpendicular of $P(2,-3,1)$ on the line
$\frac{x+1}{2}=\frac{y-3}{3}=\frac{z-2}{-1}$ be $Q.$
If direction ratios of the line segment joining P and Q be $l,m,n$, then which of the following relations are correct ?

1. $2l+3m+n=1$

2. $2l+3m-n=0$

3. $\frac{l}{2}=\frac{m}{3}=\frac{n}{-1}$

4. $\frac{l}{2}=\frac{m}{3}=\frac{n}{1}$

Q. The equation of the plane through intersection of planes $x+2y+3z=4$ and $2x+y-z=-5,$ and perpendicular to the plane $5x+3y+6z+8=0$ is

1. $7x-2y+3z+81=0$
2. $23x+14y-9z+48=0$
3. $51x+15y-50z+173=0$
4. $5x-17y+2z+23=0$

Q. Find the vector and cartesian forms of the equation of the plane passing through the point $(1,2,4)$ and parallel to the lines
$\overrightarrow{r}=\hat{i}+2\hat{j}-4\hat{k}+\lambda (2\hat{i}+3\hat{j}+6\hat{k})$
$\overrightarrow{r}=\hat{i}-3\hat{j}+5\hat{k}+\mu (\hat{i}+\hat{j}-\hat{k})$
Also, the distance of the point $(9,8,10)$ from the plane is $\frac{a}{\sqrt{b}}.$ Find the value of $a+b$

1. $156$
2. $160$
3. $170$
4. $176$

Q. $\begin{array}{cccc}& \text{List-1}& & \text{List-2}\\ \left(P\right)& \text{Let L be the line}\frac{x-6}{2}=\frac{y-12}{4}=\frac{z-8}{2}& & 1.5\\ & \text{Let A be the point of intersection of the}& & \\ & \text{plane}{p}_1:x+y+z=6\text{and}L.\text{Let P be the}& & \\ & \text{point of intersection of the plane}{p}_2:x+y+z=10& & 2.8\\ & \text{L. AP is produced to Q such that AP=PQ}& & \\ & \text{and if coordinates of Q are}(\alpha ,p,r)then& & 3.6\\ & \text{distnce of Q from xy-plane is}& & \\ \left(q\right)& \text{DABC is a tetrahedran A(2, 0, 0), B(0, 4, 0})& & 4.2\\ & \text{Edge CD lies on the line}\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}.\text{If}& & 5.0\\ & \text{locus of centroid of tetrahedron lies on}& & \\ & \frac{x-x_1}{1}=\frac{y-y_1}{a}=\frac{z-z_1}{b}\text{then a+b is}& & \\ \left(R\right)& \text{P is the area of region of complex plane}& & \\ & \text{generated by}(z:\frac{z}{4}\text{and}\frac{4}{\overline{z}}\text{have real and}& & \\ & \text{imaginary part in (0, 1) )then [p]is}& & \\ & \text{where [, ] denotes greatest integer function}& & \\ \left(S\right)& \text{If}{Z}_1=a_1+i{b}_{1}and{z}_2=a_2+i{b}_2\text{are complex}& & \\ & \text{numbers such that}|z_1|=1,|z_2|=2\text{and}& & \\ & Re\left(z_1{z}_2\right)=0\text{and let}{w}_1=a_1+\frac{i{a}_2}{2}\text{and}& & \\ & w_2=2b_1+i{b}_2\text{then}|\left(w_1{w}_2\right)|is& & \end{array}$

1. $p\to 1,q\to 3,r\to 4,s\to 5$
2. $p\to 1,q\to 1,r\to 3,s\to 5$
   
3. $p\to 4,q\to 1,r\to 1,s\to 4$
4. $p\to 1,q\to 1,r\to 1,s\to 5$
   

Q.

A point is on the x-axis. What are its y-coordinates and z-coordinates?

Q. The point collinear with $(4,2,0)$ and $(6,4,6)$ among the following is

1. $(0,4,6)$
2. $(8,6,8)$
3. $(1,-4,-6)$
4. None of these
   

Q. On the set of all points in a plane, the relation defined by the phrase 'at the same distance from the origin ' is an equivalence relation.

1. True
2. False

Q.

In each of the following exercise, determine the direction cosines of the normal to plane and the distance from the origin:

z=2

x + y + z = 1

2x +3y -z = 5

5y + 8 = 0

Q. For every point P (x, y, z) on the x-axis (except the origin),
(a) x = 0, y = 0, z ≠ 0
(b) x = 0, z = 0, y ≠ 0
(c) y = 0, z = 0, x ≠ 0
(d) x = y = z = 0

Q.

The co-ordinates of the foot of perpendicular drawn from point P(1, 0, 3)to the join of points A(4, 7, 1)and B(3, 5, 3)is
[RPET 2001]

1. 

2. 

3. 

4. 

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

Q. If the perpendicular distance of a point other than the origin from the plane x+y+z=p is equal to the distance of the plane from the origin, then the coordinates of the point are

1. (p, 2 p, 0)
2. (0, 2p, -p)
3. (2p, p, -p)
4. (2p, -p, 2p)

Q. The foot of the perpendicular from (0, 2, 3) to the line $\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}$ is:

1. (2, –1, 3)
2. (2, 3, –1)
3. (3, 2, –1)
4. (–2, 3, 4)

Q. A line is drawn in the direction of $(\hat{i}+\hat{j}-2\hat{k})$ and it passes through a point with the position vector $(2\hat{i}-\hat{j}-4\hat{k})$. Find the equations of the line in the vector as well as Cartesian forms.

Q.

If foot of the perpendicular of P(2, -3, 1) on the line $\frac{x+1}{2}=\frac{y-3}{3}=\frac{z+2}{-1}$ is Q(a, b, c) then find the value of -14(a+b+c)

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Q. Find the image of the point P (3, 5, 7) in the plane 2x+y+z = 0 ?

1. (-3, 2, 4)
2. (-9, -1, 1)
3. (-1, -9, 1)
4. None of these

Q.

If foot of the perpendicular of P(2, -3, 1) on the line $\frac{x+1}{2}=\frac{y-3}{3}=\frac{z+2}{-1}$ is Q(a, b, c) then find the value of 14(a+b+c)

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