Coordinates of a Point in Space

Q.

The points (1, 7), (4, 2), (-1, -1), (-4, 4) are the vertices of a _____

1. Parallelogram

2. Square

3. Rhombus

4. None of these

Q. The equation of sphere passing through points $(1,0,0),(0,1,0)$ and $(0,0,1)$ and having minimum radius, is

1. $3(x^2+y^2+z^2)-x-y-z-1=0$
2. $3(x^2+y^2+z^2)-2(x+y+z)-1=0$
3. $3(x^2+y^2+z^2)+2(x+y+z)+1=0$
4. $3(x^2+y^2+z^2)+x+y+z+1=0$

Q. If three planes $P_1\equiv 2x+y+z-1=0,P_2\equiv x-y+z-2=0$ and $P_3\equiv \alpha x-y+3z-5=0$ intersect each other at point $P$ on $XOY$ plane and at point $Q$ on $YOZ$ plane, where $O$ is the origin, then identify the correct statement(s)?

1. The length of projection of $\overrightarrow{PQ}$ on $x-$axis is $1.$ unit
2. Straight line perpendicular to plane $P_3$ and passing through $P$ is $\frac{x-1}{4}=\frac{y+1}{-1}=\frac{z}{3}.$
3. Centroid of the triangle $OPQ$ is $(\frac{1}{3},\frac{-1}{2},\frac{1}{2})$
4. The value of $\alpha$ is $4.$

Q.

The centre of sphere passes through four points (0, 0, 0), (0, 2, 0), (1, 0, 0) and (0, 0, 4) is

1. 
   

2. 
   

3. 
   

4. 
   

Q. Let A, B, C be the feet of perpendiculars drawn from P(3, 4, 5) on XY, YZ, ZX planes respectively, then Coordinates of A, B and C will be .

1. $(5,4,0),(0,4,3),(3,0,5)$
2. $(3,4,0),(0,3,5),(4,0,5)$
3. $(3,4,0),(0,4,5),(3,0,5)$
4. $(3,5,0),(0,4,5),(3,0,4)$

Q.

Perpendicular distance of P(x, y, z ) from XY, YZ and XZ planes respectively are 1, 2 and 3. Which of the following could be the coordinates of P?

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

2. (-1, 2 , -3)

3. (-2, 3, 1)

4. (2, -3, 1)

Q.

A cube of side 5 has one vertex at the point (1, 0, -1), and the three edges from this vertex are, respectively, parallel to the negative x and y axes and positive z-axis. Find the coordinates of the other vertices of the cube.

Q.

What is K in vertex form?

Q. Which face is opposite to face with letter $B,$ if four positions of a die are given below as :

1. $F$
2. $D$
3. $A$
4. $E$

Q.

In the given figure of cuboid if the coordinates of point E is (3, 2 , 1) and one of the corners as the origin.

How many of the following coordinates are correct?

A(0, 1, 0) B(3, 0, 1) C(3, 0, 0) D(2, 3, 0)

E(3, 2, 1) F(0, 2, 0) G(0, 2, 1) H(0, 0, 0)

Q. A particle starts moving from a point P on the curve | z - 3 - 4i| = 5, where |z| is maximum. From P it moves through an angle $ta{n}^{-1}\left(\frac{3}{4}\right)$ in in anticlockwise direction and reaches the point Q. From Q it comes down parallel to imaginary axis by 2 units and reaches the point R. Then

1. $R=(3,9)$
2. $R=(3,7)$
3. Area of $\Delta$ PQR = 3 sq. Units
4. $\Delta$ PQR is right angled

Q.

A line segment joining (1, 0, 1) and the origin (0, 0, 0) is revolved about the x-axis to form a right circular cone. If (x, y, z) is any point on the cone other than the origin, then it satisfies the equation

1. $x^2-2y^2-z^2=0$

2. $x^2-y^2-z^2=0$

3. $2x^2-y^2-2z^2=0$

4. $x^2-2y^2-2z^2=0$

Q. In a triangle $ABC,$ one of the vertex $A$ is $(3,4)$ and vertices $B(x_1,y_1),C(x_2,y_2)$ lie on the line $3x+4y=5$. The area of the triangle $ABC$ is $20$ sq. units. If $x_1+x_2=10$ and $\frac{x_1+y_1}{x_2+y_2}=\frac{3}{k},$ then the smaller value of $\left[k\right]$, (where [.] is greatest integer function) is

Q. The equation of sphere passing through $(1,0,0),(0,1,0)$ and $(0,0,1)$ and whose centre lies on the plane $3x-y+z=2$ is

1. $(x^2+y^2+z^2)-3(x+y+z)+1=0$
2. $2(x^2+y^2+z^2)-3(x+y+z)+1=0$
3. $3(x^2+y^2+z^2)-2(x+y+z)+1=0$
4. $4(x^2+y^2+z^2)-2(x+y+z)+1=0$

Q.

How many of the following statements are correct?
1. If a point lies on the y-axis its x & z-coordinates are zero.
2. If a point lies on the x-z plane, its y-coordinate is zero.

3. The x-axis and y-axis taken together to determine a plane known as x-y plane.

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Q. Name the octant in which the point lie: $(4,-3,5)$

Q.

The coordinates of a point are (3, -2, -5). Write down the coordinates of seven points such that the absolute values of their coordinates are the same as those of the coordinates of the given point.

Q.

Using section formula, show that the points A (2, –3, 4), B (–1, 2, 1) and are collinear.

Q. A rectangular parallelopiped is formed by planes drawn through the points (5, 7, 9) and (2, 3, 7) parallel to the coordinate planes.The length of the edge of the rectangular parallelopiped is

Q.

In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.

(a) (b)

(c) (d)

Q.

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse

Q. The coordinates of the point P are given by (2, -4, 5). Then,

1. Distance of point P from the XY plane = 5
2. Distance of point P from the XZ plane = $\sqrt{29}$
   
3. Distance of point P from the YZ plane = 2
   

4. Distance of point P from the XZ plane = 4

Q.

A point is in the XZ-plane. What can you say about its y-coordinate?

Q.

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse

Q. The co-ordinates of foot of the perpendicular drawn on line $3x-4y-5=0$ from the point $(0,5)$ is

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

Q.

How do I find the equation of a sphere that passes through the origin and whose center is $(4,1,2)$?

Q. The length, width and height of a cuboid are 3, 2 and 1 unit along x, y and z axes. If one corner is the origin, then cannot be a corner of the cuboid.

1. (3, 0, 1)
2. (0, 2, 0)
3. (0, 0, 0)
4. (0, 2, 1)
5. (3, 0, 0)
6. (0, 1, 0)

Q.

How many of these are correctly matched ?

Coordinates of a point Name of the octant

1. $(1,2,3)P.I$

2. $(4,-2,3)Q.IV$

3. $(4,-2,-5)R.VIII$

4. $(4,2,-5)S.V$

5. $(-4,2,-5)T.VI$

6. $(-4,2,5)U.II$

7. $(-3,-1,6)V.III$

8. $(2,-4,-7)W.VIII$

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Q. The equation of a line which is parallel to the angle bisector between the lines $\frac{x-2}{1}=\frac{y-3}{1}=\frac{z-4}{0}$ and $\frac{x-1}{0}=\frac{y-4}{-1}=\frac{z-5}{-1}$ and passing through the point $(1,-2,3)$ is

1. $\frac{x-1}{1}=\frac{y+2}{0}=\frac{z-3}{-1}$
2. $\frac{x-1}{1}=\frac{y+2}{2}=\frac{z-3}{-1}$
3. $\frac{x-1}{1}=\frac{y+2}{0}=\frac{z-3}{1}$
4. $\frac{x-1}{1}=\frac{y+2}{2}=\frac{z-3}{1}$

Q.

A line segment joining (1, 0, 1) and the origin (0, 0, 0) is revolved about the x-axis to form a right circular cone. If (x, y, z) is any point on the cone other than the origin, then it satisfies the equation

1. $x^2-2y^2-z^2=0$

2. $x^2-y^2-z^2=0$

3. $x^2-2y^2-2z^2=0$

4. $2x^2-y^2-2z^2=0$