In what ratio does the point 2, 3 divides the line joining the points 1 0 and 4 5 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] internally in the ratio m1 : m2 i.e.
PR/RQ = m1 / m2, where m1

0, m2

0, m1 + m2

Table of Contents Show

When the Point divides the line segment Externally
Mid-point formula
Illustrative Examples

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

0, m2

0, m1 - m2

0
Then the co-ordinates of R are m1 x2 -m2 x1)/(m1 -m2), (m1y2 -m2y1)/(m1 -m2)

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

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)
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

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.
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.
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.
Find the co-ordinates of the points of trisection of the line segment joining the points (3, - 1) and (-6, 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.
Find the point which is one-third of the way from P (3, 1) to Q (-2, 5).
Find the point which is two third of the way from P(0, 1) to Q(1, 0).
Find the co-ordinates of the point which is three fifth of the way from (4, 5) to (-1, 0).
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.
In what ratio does the point P (2, -5) divide the line segment joining the points A (- 3, 5) and B (4, -9)?
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.
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.
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.
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.
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.
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.
By using section formula, prove that the points (0, 3), (6, 0) and (4, 1) are collinear.
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.
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.
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.
Show that the line segments joining the points (1, - 2), (1, 2) and (3, 0), (-1, 0) bisect each other.
Show that the points A(-2, -1), B (1, 0), C (4, 3) and D (1, 2) from a parallelogram. Is it a rectangle?
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?
Three consecutive vertices of a parallelogram are (4, - 11), (5, 3) and (2, 15). Find the fourth vertex.

Answers

1. (i) (3, 1)      (ii) (-1, 9)           2. (i) (12/5, 13/5)    (ii) (0, - 7)
3. (i) (4, 8)    (ii) (1, - 2)           4. (0, 1) and (-3, 3)
5. (-1/3, 2/3) and (9, 10)          6. (4/3, 7/3)
7. (2/3, 1/3)                               8. (1, 2)
9. (4, -11)                                 10. 5 : 2 internally
11. 1 : 2 internally; (3, 0)            12. 5 : 2 externally; (-1 , 0)
13. 3 : 2 internally                      14. (- 14, 6)
15. (3, 6)                                    16. 3 : 2 internally; (2, 3)
18. (11 , 10)                               19. a = 2, b = 2
20. (-7 , 3)                                 22. No
23. Parallelogram                        24. (1, 1)

Answer

Verified

Hint: suppose, the ratio in all cases lying on the line joining given as \[k:1\]. Use sectional formula given for calculating a point which divides the line segment joining the points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$in the ratio \[m:n\]; point given as \[\left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n} \right)\] any point on x-axis has y-coordinates as 0 and vice-versa is also true. Use this logic to solve the problem. Complete step-by-step answer:We know the point which divides the line joining the points $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ in ratio of \[m:n\], is given by sectional formula as:- \[\text{R}\ =\ \left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n} \right)\] …………………………………………(i)Now, coming to the question, we need to find the ratio by which line joining $\text{A}\left( 8,9 \right)$ and $\text{B}\left( -7,4 \right)$ would be divided by the given points in the axis. (a). The point $\left( 2,7 \right)$Let us suppose $\left( 2,7 \right)$ divides the line joining $\left( 8,9 \right)$and $\left( -7,4 \right)$ in ratio of \[k:1\].

Now, we can get coordinates of c with the help of equation (i), where\[m=k\], \[n=1\] and \[\left( {{x}_{1}},{{y}_{1}} \right)\ =\ \left( 8,9 \right)\], \[\left( {{x}_{2}},{{y}_{2}} \right)\ =\ \left( -7,4 \right)\] So, we get coordinated of c as \[c\ =\ \left( \dfrac{-7k+8}{k+1},\dfrac{4k+9}{k+1} \right)\]Now, it is given that coordinates of point c is $\left( 2,7 \right)$, So, we get,\[\dfrac{-7k+8}{k+1}\ =\ 2\] and \[\dfrac{4k+9}{k+1}\ =\ 7\] \[-7k+8\ =\ 2k+2\] and \[4k+9\ =\ 7k+7\] \[9k\ =\ 6\] and \[3k\ =\ 2\] \[k\ =\ \dfrac{6}{9}\ =\ \dfrac{2}{3}\] and \[k\ =\ \dfrac{2}{3}\] Hence, ratio \[k:1\]is given as \[2:3\]. So, point $\left( 2,7 \right)$ will divide the line joining the given points in ratio of \[2:3\]. (b). The x- axis Let us suppose that any coordinate on the x-axis will divide the line joining the given points in ratio \[k:1\].

Let us suppose the point on the x-axis is represented by ‘c’. So, coordinates of c can be given with the help of equation (i) as \[c\ =\ \left( \dfrac{-7k+8}{k+1},\dfrac{4k+9}{k+1} \right)\] As, the point c is lying on the x-axis, so y-coordinate of this point should be 0 because y-coordinate of any point at x-axis is 0. So, put the y-coordinate of point c to 0, to get the value of k. So, we get \[\dfrac{4k+9}{k+1}\ =\ 0\] Or \[4k+9\ =\ 0\]\[k\ =\ -\dfrac{9}{4}\]Hence, line joining by the point given points will be divided by x-axis in ratio of \[9:4\] externally as the value of k is negative.(c). The y-axis So, we can use the previous coordinate of ‘c’. and put the x-coordinate of point c to 0, as x-coordinate on y-axis will be 0. Hence, we get \[\dfrac{-7k+8}{k+1}\ =\ 0\]\[-7k+8\ =\ 0\]\[7k\ =\ 8\]\[k\ =\ \dfrac{8}{7}\]So, the y-axis will divide the line joining the given points in ratio of \[8:7\]. Note: please take care with the positions of $\left( {{x}_{1}},{{y}_{1}} \right)$, $\left( {{x}_{2}},{{y}_{2}} \right)$ and m and n in the sectional formula. One may go wrong if he/she applies this formula as \[\left( \dfrac{m{{x}_{2}}+n{{x}_{1}}}{m+n},\dfrac{m{{y}_{2}}+n{{y}_{1}}}{m+n} \right)\] using the concept that x-coordinate of any point on y-axis is 0 and y-coordinate of any point on x-axis is 0 are the key points with the second and third party of the question. Negative value of k suggests that the point dividing it in \[k:1\] will not lie in between the line segments, it will divide the line externally, not internally.