NCERT Solutions for Class 11 Maths Chapter 12 – Introduction to Three Dimensional Geometry Ex 12.3
Page No 277:
Question 1:
Find the coordinates of the point which divides the line segment joining the points (–2, 3, 5) and (1, –4, 6) in the ratio (i) 2:3 internally, (ii) 2:3 externally.
Answer:
(i) The coordinates of point R that divides the line segment joining points P (x1, y1, z1) and Q (x2, y2, z2) internally in the ratio m: n are
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/172/4858/NCERT_16-10-08_Khushboo_11_Math_Ex-12.1_4_SU_SNK_html_374112b3.gif)
Let R (x, y, z) be the point that divides the line segment joining points(–2, 3, 5) and (1, –4, 6) internally in the ratio 2:3
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/172/4858/NCERT_16-10-08_Khushboo_11_Math_Ex-12.1_4_SU_SNK_html_m62f519fd.gif)
Thus, the coordinates of the required point are
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/172/4858/NCERT_16-10-08_Khushboo_11_Math_Ex-12.1_4_SU_SNK_html_47dd6e50.gif)
(ii) The coordinates of point R that divides the line segment joining points P (x1, y1, z1) and Q (x2, y2, z2) externally in the ratio m: n are
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/172/4858/NCERT_16-10-08_Khushboo_11_Math_Ex-12.1_4_SU_SNK_html_4dfcab66.gif)
Let R (x, y, z) be the point that divides the line segment joining points(–2, 3, 5) and (1, –4, 6) externally in the ratio 2:3
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/172/4858/NCERT_16-10-08_Khushboo_11_Math_Ex-12.1_4_SU_SNK_html_m69dbd1e.gif)
Thus, the coordinates of the required point are (–8, 17, 3).
Question 2:
Given that P (3, 2, –4), Q (5, 4, –6) and R (9, 8, –10) are collinear. Find the ratio in which Q divides PR.
Answer:
Let point Q (5, 4, –6) divide the line segment joining points P (3, 2, –4) and R (9, 8, –10) in the ratio k:1.
Therefore, by section formula,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/172/4859/NCERT_16-10-08_Khushboo_11_Math_Ex-12.1_4_SU_SNK_html_m212aed78.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/172/4859/NCERT_16-10-08_Khushboo_11_Math_Ex-12.1_4_SU_SNK_html_31cf4e57.gif)
Thus, point Q divides PR in the ratio 1:2.
Question 3:
Find the ratio in which the YZ-plane divides the line segment formed by joining the points (–2, 4, 7) and (3, –5, 8).
Answer:
Let the YZ planedivide the line segment joining points (–2, 4, 7) and (3, –5, 8) in the ratio k:1.
Hence, by section formula, the coordinates of point of intersection are given by![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/172/4860/NCERT_16-10-08_Khushboo_11_Math_Ex-12.1_4_SU_SNK_html_16a55a58.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/172/4860/NCERT_16-10-08_Khushboo_11_Math_Ex-12.1_4_SU_SNK_html_16a55a58.gif)
On the YZ plane, the x-coordinate of any point is zero.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/172/4860/NCERT_16-10-08_Khushboo_11_Math_Ex-12.1_4_SU_SNK_html_54264cf4.gif)
Thus, the YZ plane divides the line segment formed by joining the given points in the ratio 2:3.
Question 4:
Using section formula, show that the points A (2, –3, 4), B (–1, 2, 1) and
are collinear.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/172/4861/NCERT_16-10-08_Khushboo_11_Math_Ex-12.1_4_SU_SNK_html_me08e7c3.gif)
Answer:
The given points are A (2, –3, 4), B (–1, 2, 1), and
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/172/4861/NCERT_16-10-08_Khushboo_11_Math_Ex-12.1_4_SU_SNK_html_me08e7c3.gif)
Let P be a point that divides AB in the ratio k:1.
Hence, by section formula, the coordinates of P are given by
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/172/4861/NCERT_16-10-08_Khushboo_11_Math_Ex-12.1_4_SU_SNK_html_m61832ad7.gif)
Now, we find the value of k at which point P coincides with point C.
By taking
, we obtain k = 2.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/172/4861/NCERT_16-10-08_Khushboo_11_Math_Ex-12.1_4_SU_SNK_html_m60970da9.gif)
For k = 2, the coordinates of point P are
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/172/4861/NCERT_16-10-08_Khushboo_11_Math_Ex-12.1_4_SU_SNK_html_5b85adff.gif)
i.e.,
is a point that divides AB externally in the ratio 2:1 and is the same as point P.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/172/4861/NCERT_16-10-08_Khushboo_11_Math_Ex-12.1_4_SU_SNK_html_me08e7c3.gif)
Hence, points A, B, and C are collinear.
Question 5:
Find the coordinates of the points which trisect the line segment joining the points P (4, 2, –6) and Q (10, –16, 6).
Answer:
Let A and B be the points that trisect the line segment joining points P (4, 2, –6) and Q (10, –16, 6)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/172/4862/NCERT_16-10-08_Khushboo_11_Math_Ex-12.1_4_SU_SNK_html_m6cdb5ca3.jpg)
Point A divides PQ in the ratio 1:2. Therefore, by section formula, the coordinates of point A are given by
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/172/4862/NCERT_16-10-08_Khushboo_11_Math_Ex-12.1_4_SU_SNK_html_m5b46a48e.gif)
Point B divides PQ in the ratio 2:1. Therefore, by section formula, the coordinates of point B are given by
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/172/4862/NCERT_16-10-08_Khushboo_11_Math_Ex-12.1_4_SU_SNK_html_3eac4aa8.gif)
Thus, (6, –4, –2) and (8, –10, 2) are the points that trisect the line segment joining points P (4, 2, –6) and Q (10, –16, 6).