Question

Find the ratio in which the point p(b,1) divides the join of A(7,-2) and B ( -5,6). Also, find the value of b.

Answer 1

Answer:

1) m : n = 3 : 5

2) Co-ordinates of P = (3/4 , 1)

Step-by-step explanation:

Given,

P = (b , 1)

A = (7 , -2)

B = (-5 , 6)

To Find :-

1) Ratio that 'P' divides line segment 'AB'.

2) Co-ordinates of 'P'.

How To Do :-

Here they given the values of Co-ordinates of 'P' , 'A' , 'B'. So we need to equate this to internal division formula . After equating the terms we need to equate the both 'y-co-ordinates'. By equating that we can find the value of the ratio that 'P' divides line segment AB. After finding the value of the ratio , again we need to substitute the value of all the Co-ordinates and the ratio in the formula to get the value of 'b'.

Formula Required :-

Section(Internal division) formula :-

$(x , y) = \left(\dfrac{mx_2+nx_1}{m+n},\dfrac{my_2+ny_1}{m+n}\right)$

Solution :-

P = (b , 1)

A = (7 , -2)

Let,

x_1 = 7 , y_1 = -2

B = (-5 , 6)

Let,

x_2 = -5 , y_2 = 6

Substituting in the formula :-

$(b , 1) = \left(\dfrac{m(-5)+n(7)}{m+n},\dfrac{m(6)+n(-2)}{m+n}\right)$

$(b , 1) = \left(\dfrac{-5m+7n}{m+n},\dfrac{6m-2n}{m+n}\right)$

Equating both y - Co-ordinates :-

1 = 6m - 2n/m + n

1(m + n) = 6m - 2n

m + n = 6m - 2n

n + 2n = 6m - m

3n = 5m

3/5 = m/n

m : n = 3 : 5

Ratio that 'P' divides line segment AB = 3:5

Substituting the ratio in the formula :-

$(b , 1) = \left(\dfrac{3(-5)+5(7)}{3+5},\dfrac{3(6)+5(-2)}{3+5}\right)$

$(b , 1) = \left(\dfrac{-15+21}{8},\dfrac{18-10}{8}\right)$

(b , 1) = (6/8 , 8/8)

(b , 1) = (3/4 , 1)

Co-ordinates of P = (b , 1) = (3/4 , 1)