What is the ratio in which the point P () divides the line segment joining the point A () and 2 5?

Text Solution

Solution : Let the required ratio be k : 1. <br> Then, by the section formula, the coordinates of P are <br> `P((4k - 3)/(k+1), (-9k+5)/(k+1))` <br> `therefore (4k-3)/(k+1) = 2 "and" (-9k+5)/(k+1) =-5 " "[because P(2, -5) "is given"]` <br> `rArr 4k-3 = 2k+2"and" -9k+5 =-5k-5` <br> `rArr 2k =5 "and" 4k = 10` <br> `rArr k = (5)/(2)` in each case. <br> So, the required ratio is `(5)/(2):1,"which is" 5:2.` <br> Hence, P divides AB in the ratio 5:2.

Solution:

Given, point P(3/4, 5/12) divides the line segment joining the points A(1/2, 3/2) and B(2, -5).

We have to find the ratio in which the point P divides the line segment AB.

By section formula,

The coordinates of the point P(x, y) which divides the line segment joining the points A (x₁ , y₁) and B (x₂ , y₂) internally in the ratio k : 1 are [(kx₂ + x₁)/(k + 1), (ky₂ + y₁)/(k + 1)]

Here, (x₁, y₁) = (1/2, 3/2) and (x₂, y₂) = (2, -5)

[(k(2) + (1/2))/(k + 1) , (k(-5) + (3/2))/(k + 1)] = (3/4, 5/12)

[(2k + (1/2))/(k + 1), (-5k + (3/2))/(k + 1)] = (3/4, 5/12)

Now, (2k + (1/2))/(k + 1) = 3/4

2k + (1/2) = 3/4(k + 1)

(4k + 1)/2 = (3k + 3)/4

4k + 1 = (3k + 3)/2

2(4k + 1) = 3k + 3

8k + 2 = 3k + 3

8k - 3k = 3 - 2

5k = 1

k = 1/5

Also, (-5k + (3/2))/(k + 1) = 5/12

-5k + (3/2) = 5/12(k + 1)

(-10k + 3)/2 = (5k + 5)/12

(-10k + 3) = (5k + 5)/6

6(-10k + 3) = 5k + 5

-60k + 18 = 5k + 5

18 - 5 = 5k + 60k

65k = 13

k = 13/65

k = 1/5

Therefore, the required ratio is 1:5

✦ Try This: Find the ratio in which point T (-1, 6) divides the line segment joining the points P (-3, 10) and Q (6, -8).

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 7

NCERT Exemplar Class 10 Maths Exercise 7.3 Problem 11

Summary:

The ratio in which the point P(3/4, 5/12) divides the line segment joining the points A (1/2, 3/2) and B (2, –5) is 1:5

☛ Related Questions:

A line divides internally in the ratio m : n

x1 = −3, x2 = 4, y1 = 5, y2 = −9

(2, −5) = `((4"m" - 3"n")/("m" + "n"), (-9"m" + 5"n")/("m" + "n"))`

`(4"m" - 3"n")/("m" + "n")`

= 2

4m – 3n = 2m + 2n

4m – 2m = 3n + 2n

2m = 5n

`"m"/"n" = 5/2`

m : n = 5 : 2

The ratio is 5 : 2.

and

`(-9"m" + 5"n")/("m" + "n")`

= −5

−9m + 5n = −5(m + n)

−9m + 5n = −5m – 5n

−9m + 5m = −5n – 5n

−4m = −10

`"m"/"n" = 10/4`

⇒ `"m"/"n" = 5/2`

m : n = 5 : 2

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