Question

40. Find two parts
41. Find the ratio in which the line segment joining the points (-3, 10) and (6,-8)
is divided by (-1, 6).
​

Answer 1

Answer:-

Given:-

( - 1 , 6) divided the line segment joining the points ( - 3 , 10) & (6 , - 8).

Let , the ratio in which the line segment divided be m : n.

Using section formula;

i.e.,

The co - ordinates of a point divided the line segment joining the points (x₁ , y₁) & (x₂ , y₂) in the ratio m : n are;

$\boxed {\sf \: (x \: ,\: y) = \bigg( \dfrac{mx_2 + nx_1}{m + n} \: \: ,\: \: \dfrac{my_2 + ny_1}{mn} \bigg)}$

Let,

• x = - 1

• y = 6

• x₁ = - 3

• y₁ = 10

• x₂ = 6

• y₂ = - 8

Hence,

$\implies \sf \: ( - 1 \: , \: 6) = \bigg( \frac{m(6) + n( - 3)}{m + n} \: \: ,\: \: \frac{m( - 8) + n(10)}{m + n} \bigg) \\ \\ \\ \implies \sf \: ( - 1 \: , \: 6) = \bigg( \frac{6m - 3n}{m + n} \: \: ,\: \: \frac{ - 8m+ 10n}{m + n} \bigg)$

On comparing both sides we get;

$\implies \sf \: - 1 = \frac{6m - 3n}{m + n} \\ \\ \\ \implies \sf \: - m - n = 6m - 3n \\ \\ \\ \implies \sf \: - n + 3n = 6m + m \\ \\ \\ \implies \sf \:2n = 7m \\ \\ \\ \implies \sf \: \frac{2}{7} \times n = m \\ \\ \\ \implies \sf \: \frac{2}{7} = \frac{m}{n} \\ \\ \\ \implies \boxed{\sf \:m :n = 2 : 7}$

∴ The ratio in which the line segment is divided is 2 : 7.

Answer 2

Answer:

Correct Question :-

• Find the ratio in which the line segment joining the points (- 3 , 10) and (6 , - 8) is divided by (- 1 , 6).

Given :-

• The line segment joining the points (- 3 , 10) and (6 , - 8) is divided by (- 1 , 6).

To Find :-

• What is the ratio of the line segment.

Formula Used :-

By using section formula we know that,

$\sf\boxed{\bold{\pink{(x , y) =\: \bigg(\dfrac{mx_2 + nx_1}{m + n} , \dfrac{my_2 + ny_1}{mn}\bigg)}}}$

Solution :-

Let, the ratio of line segment be m : n.

Given :

• x = - 1
• x₁ = - 3
• x₂ = 6
• y = 6
• y₁ = 10
• y₂ = - 8

According to the question by using the formula we get,

$\implies \sf (- 1 , 6) =\: \bigg(\dfrac{m(6) + n(- 3)}{m + n} , \dfrac{m(- 8) + n(10)}{mn}\bigg)$

$\implies \sf (- 1 , 6) =\: \bigg(\dfrac{m \times 6 + n \times (- 3)}{m + n)} , \dfrac{m \times (- 8) + n \times 10}{mn}\bigg)$

$\implies \sf (- 1 , 6) =\: \bigg(\dfrac{6m - 3n}{m + n} , \dfrac{- 8m + 10n}{mn}\bigg)$

$\implies \sf \dfrac{6m - 3n}{m + n} =\: - 1$

By doing cross multiplication we get,

$\implies \sf 6m - 3n =\: - (m + n)$

$\implies \sf 6m - 3n =\: - m - n$

$\implies \sf 6m + m =\: - n + 3n$

$\implies \sf 7m =\: 2n$

Then, we can write as :

$\implies \sf \dfrac{m}{n} =\: \dfrac{2}{7}$

$\implies \sf\bold{\red{m : n =\: 2 : 7}}$

$\therefore$ The ratio of the line segment is 2 : 7.