**Exercise 12.3**

** ****Q.1: Find the coordinates of the point which divides the line segment joining the points (-2, 1, 0) and (1, 3, 6) internally in the ratio of 1:4 and externally in the ratio of 1:4.**

**Sol:**

**Since, the coordinates of point P, which divides the line segment joining two points A (a _{1}, b_{1}, c_{1}) and B (a_{2}, b_{2}, c_{2}) internally in the ratio m and n are:**

Therefore, the **coordinates** of **point P**, which divides the line segment joining two points **A(-2, 1, 0) **and **B (1, 3, 6) internally** in the ratio **1 and 4** are:

**(x, y, z) =**

**(x, y, z) = **

**Therefore, the coordinates of point P are: **

**Now, The coordinates of point Q, which divides the line segment joining two points A (a _{1}, b_{1}, c_{1}) and B (a_{2}, b_{2}, c_{2}) Externally in the ratio m and n are:**

Therefore, the **coordinates** of **point Q**, which divides the line segment joining two points **A(-2, 1, 0) **and **B (1, 3, 6) externally **in the ratio **1 and 4** are:

**(x, y, z) =**

**(x, y, z) =**

**Therefore, the coordinates of point Q are: [−3,13,−2]**

** **** **

**Q.2: Given that the points A (3, 2, -4) B (5, 4, -6) and C (9, 8, -10), are collinear. Find the ratio in which line AC is divided by B.**

**Sol.**

Since, **point A, point B and point C** are **collinear. **Therefore, let us assume that **point B** **divides** the line segment joining **point A** and **point C **in the **ratio k : 1.**

**Hence, by Section Formula, **coordinates of **point B** are:

And, the **coordinates** of **point B** are: **(5, 4, -6) [Given]**

Therefore, on comparing it with **equation (1)** we will get:

Therefore, k =

**Hence, point B divides AB in the ratio of 1 : 2**

** **

**Q.3: Find the ratio in which XZ – plane divides the line segment AB formed by joining the points (3, -5, 8) and (-2, 4, 7).**

**Sol.**

**Let the XZ plane divides the line segment PQ in the ratio k : 1**

**Hence, by Section Formula the coordinates of point of intersection are:**

Now, in XZ – plane, the y – coordinate of any point is zero.

**Therefore, k=54**

**Hence, the XZ plane divides the line segment formed by the joining of given points in the ratio of 2 : 3.**

**Q.4: By using the section formula; Show that the points P (-1, 2, 1), Q (0, 13, 2) and R (2, -3, 4) are collinear.**

**Sol. **

**Let, point A divides line PQ in the ratio k : 1**

**Hence, by Section Formula the coordinates of point A are:**

Therefore, the **coordinates** **of point A** are:

Now, the value of **‘k’** for which **point A** coincides with **point R**:

Therefore, **k =**

Now, for **k =** **point A** are:

**Therefore**, **the** **coordinates** of **point A** are: **(2, -3, 4)**

Hence, the coordinates of **point A** coincides with the coordinates of point R.

**R (2, -3, 4)** is a point that divides **PQ** **externally** and is same as **point A.**

**Hence, the points P, Q and R are colinear.**

**Q.5: The line segment joining points A (5, 3, -6) and B (9, 15, 7) is trisected by the points P and Q, Find the coordinates of points P and Q.**

**Sol.**** **

**Point P divides** the **line segment AB** in the **ratio 1 : 2** and **point Q** **divides** the **line segment AB** in the **ratio 2 : 1.**

Therefore, by **section formula** the coordinates of **point P** and **point Q** are:

**For Point P:**

**Therefore, the coordinates of point P are: [193,7,−53]**

**For Point Q:**

**Therefore, the coordinates of point Q are: [233,11,83]**