In which octant do the points lie?
(i) 1st octant
(ii) 4th octant
(iii) viiith octant
(iv) vth octant
(v) 2nd octant
(vi) 3rd octant
(vii) viiith octant
(viii) vith octant
Find the coordinates of the feet of perpendiculars from a point P on the xy, yz and zx-planes respectively for the given cases.
(i) (3,0,0), (0,4,0), (0,0,2)
(ii) (−5,0,0), (0,3,0), (0,0,7)
(iii) (4,0,0), (0,−3,0), (0,0,5)
Find how far apart the points (2, 0, 0) and (−3, 0, 0) are.
5
Find the distance from the origin to the point (6, 6, 7).
11
Show that if \(x^2 + y^2 = 1\), then the point \((x, y, \sqrt{1 - x^2 - y^2})\) is at a distance 1 unit from the origin.
1
Show that the points A (1, −1, 3), B (2, −4, 5) and (5, −13, 11) are collinear.
(2, −4, 16)
Three consecutive vertices of a parallelogram ABCD are A (6, −2, 4), B (2, −4, −8), and C (−2, 2, 4). Find the coordinates of the fourth vertex.
(−2, −2, −1)
Show that the triangle ABC with vertices A (0, 4, 1), B (2, 3, −1) and C (4, 5, 0) is right angled.
(−3, −4, −7)
(7, 2, 5)
(−3, 12, 17)
Find the third vertex of a triangle whose centroid is origin and two vertices are (2, 4, 6) and (0, −2, −5).
(4, 7, 6)
Find the centroid of a triangle the mid-point of whose sides are given as D (1, 2, −3), E (3, 0, 1), and F (−1, 1, −4).
(4, −5, 1)
The mid-points of the sides of a triangle are (5, 7, 11), (0, 8, 5), and (2, 3, −1). Find its vertices.
(3, −2, −1)
Three vertices of a parallelogram ABCD are A (1, 2, 3), B (−1, −2, −1), and C (2, 3, 2). Find the fourth vertex D.
(7/2, 13/2, 9)
Find the coordinates of the points which trisect the line segment joining A (2, 1, −3) and B (5, −8, 3).
2:1 externally
If the origin is the centroid of a triangle ABC having vertices A (a, 1, 3), B (−2, b, −5), and C (4, 7, c), find the values of a, b, c.
a = −2, b = −8, c = 2
Let A (2, 2, −3), B (5, 6, 9) and C (2, 7, 9) be the vertices of a triangle. The internal bisector of angle A meets BC at point D. Find the coordinates of D.
(7/2, 13/2, 9)
Show that the three points A (2, 3, 4), B (−1, 2, −3) and C (−4, 1, −10) are collinear and find the ratio in which C divides AB.
2:1 externally
The distance of point P (3, 4, 5) from the yz-plane is?
(7/2, 13/2, 9)
Show that the three points A (2, 3, 4), B (−1, 2, −3) and C (−4, 1, −10) are collinear and find the ratio in which C divides AB.
2:1 externally
The mid-point of the sides of a triangle are (1, 5, −1), (0, 4, −2) and (2, 3, 4). Find its vertices. Also find the centroid of the triangle.
vertices are (3,4,5), (−1,6,−7), (1,2,3) and centroid is (1,4, 1/3)
Prove that the points (0, −1, −7), (2, 1, −9) and (6, 5, −13) are collinear. Find the ratio in which the first point divides the join of the other two.
1:3 externally
What are the coordinates of the vertices of a cube whose edge is 2 units, one of whose vertices coincides with the origin and the three edges passing through the origin coincide with the positive direction of the axes through the origin?
(2,0,0), (2,2,0), (0,2,0), (0,2,2), (2,0,2), (0,0,0), (2,2,2)
The distance of point P(3, 4, 5) from the yz-plane is
3 units
4 units
5 units
550
What is the length of foot of perpendicular drawn from the point P(3, 4, 5) on y-axis
\(\sqrt{41}\)
\(\sqrt{34}\)
5
none of these
Distance of the point (3, 4, 5) from the origin (0, 0, 0) is
\(\sqrt{50}\)
3
4
5
If the distance between the points \((a,0,1)\) and \((0,1,2)\) is \(\sqrt{27}\), then the value of \(a\) is
5
\(\pm 5\)
-5
none of these
\(x\)-axis is the intersection of two planes
xy and xz
yz and zx
xy and yz
none of these
Equation of y-axis is considered as
\(x = 0,\; y = 0\)
\(y = 0,\; z = 0\)
\(z = 0,\; x = 0\)
none of these
The point \((-2, -3, -4)\) lies in the
First octant
Seventh octant
Second octant
Eighth octant
A plane is parallel to yz-plane so it is perpendicular to :
x-axis
y-axis
z-axis
none of these
The locus of a point for which \(y = 0,\; z = 0\) is
equation of x-axis
equation of y-axis
equation of z-axis
none of these
The locus of a point for which \(x = 0\) is
xy-plane
yz-plane
zx-plane
none of these
If a parallelepiped is formed by planes drawn through the points \((5,8,10)\) and \((3,6,8)\) parallel to the coordinate planes, then the length of diagonal of the parallelepiped is
\(2\sqrt{3}\)
\(3\sqrt{2}\)
\(\sqrt{2}\)
\(\sqrt{3}\)
\(L\) is the foot of the perpendicular drawn from a point \(P(3,4,5)\) on the xy-plane. The coordinates of point \(L\) are
(3, 0, 0)
(0, 4, 5)
(3, 0, 5)
none of these
\(L\) is the foot of the perpendicular drawn from the point \((3,4,5)\) on x-axis. The coordinates of \(L\) are
(3, 0, 0)
(0, 4, 0)
(0, 0, 5)
none of these
The three axes OX, OY, OZ determine ____.
Three coordinates planes
The three planes determine a rectangular parallelepiped which has ______ of rectangular faces.
Three pairs
The coordinates of a point are the perpendicular distance from the ______ on the respective axes.
given point
The three coordinate planes divide the space into ______ parts.
Eight
If a point P lies in yz-plane, then the coordinates of a point on yz-plane is of the form ______.
(0, y, z)
The equation of yz-plane is ______.
x = 0
If the point P lies on z-axis, then coordinates of P are of the form ______.
(0, 0, z)
The equation of z-axis, are ______.
x = 0 , y = 0
A line is parallel to xy-plane if all the points on the line have equal ______.
z- coordinates
A line is parallel to x-axis if all the points on the line have equal ______.
(y, z coordinates)
x = a represent a plane parallel to ______.
yz-plane
The plane parallel to yz-plane is perpendicular to ______.
x-axis
The length of the longest piece of a string that can be stretched straight in a rectangular room whose dimensions are 10, 13 and 8 units are ______.
\(\sqrt{333}\)
If the distance between the points (a, 2, 1) and (1, –1, 1) is 5, then a ______.
a = 5 or -3
If the mid-points of the sides of a triangle AB; BC; CA are D (1, 2, –3), E (3, 0, 1) and F (–1, 1, –4), then the centroid of the triangle ABC is ______.
(1, 1, -2)
Match the items in Column A with Column B using the table below.
| Column A | Column B |
|---|---|
(a) In xy-plane | (i) Ist octant |
(b) Point (2, 3, 4) lies in the | (ii) yz-plane |
(c) Locus of the points having x coordinate 0 is | (iii) z-coordinate is zero |
(d) A line is parallel to x-axis if and only if | (iv) z-axis |
(e) If x = 0, y = 0 taken together will represent the | (v) plane parallel to xy-plane |
(f) z = c represent the plane | (vi) if all the points on the line have equal y and z-coordinates |
(g) Planes x = a, y = b represent the line | (vii) from the point on the respective |
(h) Coordinates of a point are the distances from the origin to the feet of perpendiculars | (viii) parallel to z-axis |
(i) A ball is the solid region in the space enclosed by a | (ix) disc |
(j) Region in the plane enclosed by a circle is known as a | (x) sphere |
| Column A | Matched Item from Column B |
|---|---|
(a) In xy-plane | (iii) z-coordinate is zero |
(b) Point (2, 3, 4) lies in the | (i) Ist octant |
(c) Locus of the points having x coordinate 0 is | (ii) yz-plane |
(d) A line is parallel to x-axis if and only if | (vi) if all the points on the line have equal y and z-coordinates |
(e) If x = 0, y = 0 taken together will represent the | (iv) z-axis |
(f) z = c represent the plane | (v) plane parallel to xy-plane |
(g) Planes x = a, y = b represent the line | (viii) parallel to z-axis |
(h) Coordinates of a point are the distances from the origin to the feet of perpendiculars | (vii) from the point on the respective |
(i) A ball is the solid region in the space enclosed by a | (x) sphere |
(j) Region in the plane enclosed by a circle is known as a | (ix) disc |