In what ratio does the point (24/11 y)

Text Solution

Solution : Let M `((24)/(11), y)` divide the line segment joining the points <br> <img src="https://d10lpgp6xz60nq.cloudfront.net/physics_images/NTN_MATH_X_C07_S01_033_S01.png" width="80%"> <br> `P(2, -2) and Q(3, 7)` in the ratio `k:1`. <br> `therefore" "(24)/(11)=(k(3)+1(2))/(k+1)" "` (by using section formula) <br> `rArr" "11(3k+2)=24(k+1) rArr" "33k+22=24k +24` <br> `rArr" "33k-24k = 24-22rArr" "9k=2` <br> `therefore" "k=(2)/(9)` <br> `therefore` Required ratio = `k:1` <br> `i.e., " "(2)/(9):1` <br> `i.e., " "2:9` internally. <br> `therefore ` Required ratio = 2: 9

Let the point P`(24/11, y)` divide the line PQ in the ratio k : 1.

Then, by the section formula:

`x = (mx_2+nx_1)/(m+n), y = (my_2 + ny_1)/(m + n)`

The coordinates of R are `(24/11, y)`

`24/11 = (3k + 2)/(k + 1), y = (7k - 2)/(k + 1)`

`=>24(k + 1) = 33k + 22, y(k + 1)= 7k - 2`

⇒24k + 24 = 33k + 22  ,   yk + y =7k − 2

⇒2 = 9k

`=> k = 2/9`

Now consider the equation yk + y = 7k - 2 and put `k = 2/9`

`=> 2/9y + y = 14/9 - 2`

`=> 11/9y = (-4)/9`

`=> y = (-4)/11`

Therefore, the point R divides the line PQ in the ratio 2 : 9

And, the coordinates of R are `(24/11, (-4)/11)`

In what ratio does the point 24/11, y divide the line segment joining the points P 2, 2 and Q 3,7 ? Also find the value of y.

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