In what ratio does the point (- 4 B divide the line segment joining the points P 2 2?

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

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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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

Let P divides the line segment AB in the ratio  k : 1
Using section formula

x = `(m_1x_2 + m_2x_1)/(m_1+m_2), y = (m_1y_2 + m_2y_1)/(m_1+m_2)`

A(-6, 10) and B(3, 8)
m1 : m2 = k : 1
plugging values in the formula we get

- 4 = `( k xx 3 + 1 xx (-6))/(k + 1), y = ( k xx (- 8) + 1 xx 10)/(k + 1)`

- 4 = `( 3k - 6)/(k + 1), y = (-8k + 10)/(k + 1)`

Considering only x coordinate to find the value of k- 4k - 4 = 3k - 6- 7k = - 2k = `2/7`k : 1 = 2 : 7Now, we have to find the value of yso, we will use section formula only in y coordinate to find the value of y.

y = `(2 xx (- 8) + 7 xx 10)/(2 + 7)`

y = `( - 16 + 70 )/(9)`

y = 6Therefore, P divides the line segment AB in 2 : 7 ratio

And value of y is 6.

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Question 52 Section Formula Exercise 11

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Answer:

Solution:

In coordinate geometry, the Section formula is used to determine the internal or external ratio at which a line segment is divided by a point.

Let the ratio that the point (-4,b) divide the line segment joining the points (2,-2) and (-14,6) be m:n,

Here x1 = 2 , y1 = -2 , x2 = -14, y2 = 6, x = -4, y = b

By section formula,

x=\frac{\left(mx_2+nx_1\right)}{(m+n)}\\-4=\frac{\left(m\times-14+n\times2\right)}{(m+n)}\\-4=\frac{\left(-14m+2n\right)}{(m+n)}\\-4(m+n)=-14m+2n\\-4m-4n=-14m+2n\\-4m+14m=2n+4n\\ 10m=6n\\ \frac{m}{n}=\frac{6}{10}=\frac{3}{5}

Hence the ratio m:n is 3:5.

By Section formula,

y=\frac{\left(my_2+ny_1\right)}{\left(m+n\right)}\\b=\frac{\left(3\times6+5\times-2\right)}{(3+5)}\\b=\frac{\left(18-10\right)}{8}\\b=\frac{8}{8}\\b=1

Hence the value of b is 1 and the ratio m:n is 3:5.

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