NCERT Solutions For Class 11 Maths Chapter 12 Exercise 12.1
Introduction to Three Dimensional Geometry
Exercise 12.1
Chapter 12 Ex.12.1 Question 1
A point is on the \(x - \)axis. What are its \(y - \)coordinate and \(z - \)coordinates?
Solution
If a point is on the \(x - \)axis, then the coordinates of \(y\) and \(z\) are \(0.\)
So, the point is \(\left( x,0,0 \right)\).
Chapter 12 Ex.12.1 Question 2
A point is in the \(XZ - \)plane. What can you say about its \(y - \)coordinate?
Solution
If a point is in \(XZ - \)plane, then its \(y - \)coordinate is \(0.\)
Chapter 12 Ex.12.1 Question 3
Name the octants in which the following points lie:
\(\begin{align}&\left( {1,2,3} \right),\;\left( {4, - 2,3} \right),\;\left( {4, - 2, - 5} \right),\\&\left( {4,2, - 5} \right),\;\left( { - 4,2, - 5} \right),\;\left( { - 4,2,5} \right),\;\\&\left( { - 3, - 1,6} \right),\left( {2, - 4, - 7} \right). \end{align}\)
Solution
Here is the table which represents the octants:
Octants |
I |
II |
III |
IV |
V |
VI |
VII |
VIII |
\(x\) |
\( + \) |
\( - \) |
\( - \) |
\( + \) |
\( + \) |
\( - \) |
\( - \) |
\( + \) |
\(y\) |
\( + \) |
\( + \) |
\( - \) |
\( - \) |
\( + \) |
\( + \) |
\( - \) |
\( - \) |
\(z\) |
\( + \) |
\( + \) |
\( + \) |
\( + \) |
\( - \) |
\( - \) |
\( - \) |
\( - \) |
(i) \(\left( {1,2,3} \right)\)
Here \(x\) is positive, \(y\) is positive, and \(z\) is positive.
So, it lies in \(I\) octant.
(ii) \(\left( {4, - 2,3} \right)\)
Here \(x\) is positive, \(y\) is negative, and \(z\) is positive.
So, it lies in IV octant.
(iii) \(\left( {4, - 2, - 5} \right)\)
Here \(x\) is positive, \(y\) is negative, and \(z\) is negative.
So, it lies in VIII octant.
(iv) \(\left( {4,2, - 5} \right)\)
Here \(x\) is positive, \(y\) is positive, and \(z\) is negative.
So, it lies in V octant.
(v) \(\left( { - 4,2, - 5} \right)\)
Here \(x\) is negative, \(y\) is positive, and \(z\) is negative.
So, it lies in VI octant.
(vi) \(\left( { - 4,2,5} \right)\)
Here \(x\) is negative, \(y\) is positive, and \(z\) is positive.
So, it lies in II octant.
(vii) \(\left( { - 3, - 1,6} \right)\)
Here \(x\) is negative, \(y\) is negative, and \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\) is positive.
So, it lies in III octant.
(viii) \(\left( {2, - 4, - 7} \right)\)
Here \(x\) is positive, \(y\) is negative, and \(z\) is negative.
So, it lies in VIII octant.
Chapter 12 Ex.12.1 Question 4
Fill in the blanks:
(i) The \(x\)-axis and \(y\)-axis taken together determine a plane known as _____________.
(ii) The coordinates of points in the \(\rm{XY}\)-plane are of the form _____________.
(iii) Coordinate planes divide the space into _____________octants..
Solution
(i) The x-axis and y-axis taken together determine a plane known as XY plane.
(ii) The coordinates of points in the \(\rm{XY}\)-plane are of the form (x, y, 0).
(iii) Coordinate planes divide the space into eight octants.