In what ratio is the line joining the points 2 3 and 4 1 divides the segment joining the points?

Let the line x-y-2=0  divide the line segment joining the points A (3,1) and B (8,9)  in the ratio k : 1 at P.

Then, the coordinates of P are

`p ((8k+3)/(k+1),(9k-1)/(k+1))`

Since, P lies on the line  x - y -2 =0 we have:

` ((8k+3)/(k+1)) - ((9k-1)/(k+1)) -2=0`

⇒ 8k + 3- 9k + 1- 2k - 2 = 0 

⇒ 8k -9k -2k +3+1 - 2 = 0

⇒ -3k +2 = 0

⇒ - 3k=-2

`⇒ k =2/3`

So, the required ratio is  `2/3:1 `which is equal to 2 : 3.

Find the coordinates of the points which divide the line segment joining A(– 2, 2) and B(2, 8) into four equal parts.

Let P, Q and R be the three points which divide the line-segment joining the points A(-2, 2) and B(2, 8) in four equal parts.

Case I. For point P, we have

Hence, m1 = 1, m2 = 3
x1 = -2, y2 = 2
x2 = 2, y2 = 8Then, coordinates of P are given by

Case II. For point Q, we have

m1 = 2, m2 = 2
x1 = -2, y1 = 2
and    x2 = 2, y2 = 8Then, coordinates of Q are given by

Case III. For point R, we have

The HTET Answer Key was released on 4th December 2022 on the official website. The Answer Keys are released from levels 1, level 2, and level 3. Candiates can challenge the answer key from 5th December 2022 to 7th December 2022 till 5:00 pm. The HTET exam was conducted on the 3rd and 4th of December 2022. This exam was an MCQ based on a total of 150 marks for each level with no negative marking. The exam is conducted by the Board of School Education, Haryana to shortlist eligible candidates for PGT and TGT posts in Government schools across Haryana.

Let P (x1, y1) and Q (x2, y2) be two given points in the co-ordinate plane, and R (x, y) be the point which divides the segment [PQ] internally in the ratio m1 : m2 i.e.
    PR/RQ = m1 / m2, where m1

Then the coordinates of R are (m1 x2 +m2 x1)/(m1 + m2), (m1y2 + m2y1)/(m1 + m2)

Note. [PQ] stands for the portion of the line PQ which is included between the points P and Q including the points P and Q. [PQ] is called segment directed from P to Q. It may be observed that [QP] is the segment directed from Q to P. If a point R divides [PQ] in the ratio m1 : m2 then it divides [QP] in the ratio m2 : m1.

When the Point divides the line segment Externally

Let P (x1, y1) and Q (x2, y2) be two given points in the co-ordinate plane, and R (x, y) be the point which divides the segment [PQ] externally in the ratio m1 : m2 i.e.
     PR/RQ = m1 / m2, where m1

Mid-point formula

The co-ordinates of the mid-point of [PQ] are ((x1 +x2)/2, (y1 +y2)/2)

Illustrative Examples

Example

Find the co-ordinates of the point which divides the line segment joining the points P (2, -3) and Q (-4, 5) in the ratio 2 : 3 (i) internally (ii) externally.

Solution

1. Let (x, y) be the co-ordinates of the point R which divides the line segment joining the points P (2, -3) and Q (-4, 5) in the ratio 2 : 3 internally, then x = [2.(-4) +3.2]/(2+3) = - 2/5 and y = [2.5 +3.(-3)]/(2+3) = 1/5

   Hence the co-ordinates of R are (-2/5, 1/5)

2. Let (x, y) be the co-ordinates of the point R which divides the line segment joining the points P (2, - 3) and Q (-4, 5) in the ratio 2 : 3 externally i.e.internally in the ratio 2 : -3.
                               
   
   x = [2.(-4) + (-3).2]/[2 +(-3)] = -14/1 = 14 and y = [2.5 + (-3)(-3)]/[2 +(-3)] = 19/(-1) -19

   Hence the co-ordinates of R are (14, -19).

Example

In what ratio is the line segment joining the points (4, 5) and (1, 2) divided by the y-axis? Also find the co-ordinates of the point of division.

Solution

Let the line segment joining the points A (4, 5) and B (1, 2) be divided by the y-axis in the ratio k : 1 at P. By section formula, co-ordinates of P are ((k +4)/(k+1), (2k +5)/(k+1)). But P lies on y-axis, therefore, x-coordinate of P = 0 =>    (k +4)/(k+1) = 0   =>     k +4 = 0      =>    k = -4 The required ratio is -4 : 1 or 4 : 1 externally. Also the co-ordinates of the point of division are

    (0, (2.(-4) +5)/(-4+1)) i.e (0, 1)

Exercise

1. Find the co-ordinates of the point which divides the join of the points (2, 3) and (5, -3) in the ratio 1 : 2 (i) internally

   (ii) externally.

2. Find the co-ordinates of the point which divides the join of the points (2, 1) and (3, 5) in the ratio 2 : 3 (i) internally

   (ii) externally.

3. Find the co-ordinates of the point that divides the segment [PQ] in the given ratio: (i) P (5, -2), Q (9, 6) and ratio 3 : 1 internally.

   (ii) P (-7, 2), Q (-1, -1) and ratio 4 : 1 externally.

4. Find the co-ordinates of the points of trisection of the line segment joining the points (3, - 1) and (-6, 5).
5. Find point (or points) on the line through A (- 5, -4) and B (2, 3) that is twice as far from A as from B.
6. Find the point which is one-third of the way from P (3, 1) to Q (-2, 5).
7. Find the point which is two third of the way from P(0, 1) to Q(1, 0).
8. Find the co-ordinates of the point which is three fifth of the way from (4, 5) to (-1, 0).
9. If P (1, 1) and Q (2, -3) are two points and R is a point on PQ produced such that PR = 3 PQ, find the co-ordinates of R.
10. In what ratio does the point P (2, -5) divide the line segment joining the points A (- 3, 5) and B (4, -9)?
11. In what ratio is the line joining the points (2, - 3) and (5, 6) divided by the x-axis? Also find the co-ordinates of the point of division.
12. In what ratio is the line joining the points (4, 5) and (1, 2) divided by the x-axis? Also find the co-ordinates of the point of division.
13. In what ratio is the line joining the points (3, 4) and (- 2, 1) divided by the y-axis? Also find the co-ordinates of the point of division.
14. Point C (-4, 1) divides the line segment joining the points A (2, - 2) and B in the ratio 3 : 5. Find the point B.
15. The point R (-1, 2) divides the line segment joining P (2, 5) and Q in the ratio 3 : 4 externally, find the point Q.
16. Find the ratio in which the point P whose ordinate is 3 divides the join of (-4, 3) and (6, 3), and hence find the co-ordinates of P.
17. By using section formula, prove that the points (0, 3), (6, 0) and (4, 1) are collinear.
18. Points P, Q, R are collinear. The co-ordinates of P, Q are (3, 4), (7, 7) respectively and length PR = 10 unit, find the co-ordinates of R.
19. The mid-point of the line segment joining (2 a, 4) and (-2, 3 b) is (1, 2 a +1). Find the values of a and b.
20. The center of a circle is (-1, 6) and one end of a diameter is (5, 9), find the co-ordinates of the other end.
21. Show that the line segments joining the points (1, - 2), (1, 2) and (3, 0), (-1, 0) bisect each other.
22. Show that the points A(-2, -1), B (1, 0), C (4, 3) and D (1, 2) from a parallelogram. Is it a rectangle?
23. The vertices of a quadrilateral are (1, 4), (- 2, 1), (0, -1) and (3, 2). Show that the diagonals bisect each other. What does quadrilateral become?
24. Three consecutive vertices of a parallelogram are (4, - 11), (5, 3) and (2, 15). Find the fourth vertex.

Answers