Question

In what ratio does the point p(-4,6) divide the line segment joining the points A(-6,10) and B(3,-8)​

Answer 1

Answer:

2: 7

Step-by-step explanation:

Given,

P = (-4 , 6)

A = (-6 , 10)

B = (3 , - 8)

To Find :-

Ratio that P divides line segment AB .

How To Do :-

We need to consider that ratio as 'm : n' as we need apply the section(internal division ) formula to find the ratio in that we need to substitute the values of the Co-ordinates P, A, B.

Formula Required :-

Section(internal division) formula :-

$(x,y)=\left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+my_1}{m+n}\right)$

Solution :-

Let the ratio be 'm : n'

P = (-4 , 6)

Let,

x = - 4 , y = 6

A = (-6 , 10)

Let,

x_1 = - 6 , y_1 = 10

B = (3 , - 8)

Let,

x_2 = 3 , y_2 = - 8

Substituting the values in the formula :-

$(-4,6)=\left(\dfrac{m(3)+n(-6)}{m+n},\dfrac{m(-8)+n(10)}{m+n}\right)$

$(-4,6)=\left(\dfrac{3m-6n}{m+n},\dfrac{-8m+10n}{m+n}\right)$

Equating both x - co-ordinates and y-co - ordinates :-

First equating x co-ordinates :-

-4 = 3m - 6n/m + n

-4(m + n) = 3m - 6n

- 4m - 4n = 3m - 6n

- 4m - 3m = - 6n + 4n

- 7m = - 2n

m/n = - 2/7

m/n = 2/7

m : n = 2 : 7

Equation y co-ordinates :-

6 = -8m + 10n/ m + n

6(m + n) = -8m + 10n

6m + 6n = - 8m + 10n

6m + 8m = 10n - 6n

14m = 4n

m/n = 4/14

m/n = 2/7

m : n = 2 : 7

We can observe that both the ratios are same .

∴ m : n = 2 : 7