Question

$\displaystyle\huge\red{\underline{\underline{QUESTION}}}$

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

Answer 1

Answer:

Consider points as A(-3,10), B(6,-8), C(-1,6)

Use section formula : \frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}

m+n

mx

2

+nx

1

,

m+n

my

2

+ny

1

Substitute the values of x_1 , x_2x

1

,x

2

so, \frac{m(-3)+n(6)}{m+n}=-1

m+n

m(−3)+n(6)

=−1

which implies, -3m+6n=-m-n−3m+6n=−m−n

-2m=-7n−2m=−7n

implies,

\frac{m}{n} = \frac{7}{2}

n

m

=

2

7

Therefore, the point divides the line in ratio 7:2

Answer 2

Given points:-

• (-3, 10), (6, -8) and (-1, 6)

To Find:-

• The ratio in which the line segment joining these points is divided by (-1, 6) or m₁ : m₂

Solution:-

Let A and B be the points on (-3, 10) and (6, -8) and P be the points on (-1, 6)

Now,

The Coordinates are as follows:-

• A(-3, 10)
• B(6, -8)
• P(-1, 6)

We have,

• x₁ = -3
• x₂ = 6
• y₁ = 10
• y₂ = -8

We know,

$\dag\underline{\boxed{\pink{\rm{Section\:Formula = \bigg(\dfrac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \dfrac{m_1 y_2 + m_2 y_1}{m_1 + m_2}\bigg)}}}}$

Putting all the values in the formula:-

$\sf{P(x, y) = \dfrac{m_1 x_2 + m_2 x_1}{m_1 + m_2}, \dfrac{m_1 y_2 + m_2 y_1}{m_1 + m_2}}$

$= \sf{P(-1, 6) = \dfrac{m_1 \times 6 + m_2 \times -3}{m_1 + m_2}, \dfrac{m_1 \times -8 + m_2 \times 10}{m_1 + m_2}}$

$= \sf{(-1, 6) = \dfrac{6m_1 + (-3m_2)}{m_1 + m_3}, \dfrac{-8m_1 + 10m_2}{m_1 + m_2}}$

For x-coordinate:-

$= \sf{-1 = \dfrac{6m_1 - 3m_2}{m_1 + m_2}}$

$= \sf{-(m_1 + m_2) = 6m_1 - 3m_2}$

$= \sf{-m_1 - m_2 = 6m_1 - 3m_2}$

$= \sf{-m_1 - 6m_1 = -3m_2 + m_2}$

$= \sf{-7m_1 = -2m_2}$

$= \sf{\dfrac{m_1}{m_2} = \dfrac{-2}{-7}}$

$\implies \sf{\dfrac{m_1}{m_2} = \dfrac{2}{7}}$

$\implies \sf{m_1 : m_2 = 2 : 7}$

Also,

For y-coordinate:-

$= \sf{6 = \dfrac{-8m_1 + 10m_2}{m_1 + m_2}}$

$= \sf{6(m_1 + m_2) = -8m_1 + 10m_2}$

$= \sf{6m_1 + 6m_2 = -8m_1 + 10m_2}$

$= \sf{6m_1 + 8m_1 = 10m_2 - 6m_2}$

$= \sf{14m_1 = 4m_2}$

$= \sf{\dfrac{m_1}{m_2} = \dfrac{4}{14}}$

$\implies \sf{\dfrac{m_1}{m_2} = \dfrac{2}{7}}$

$\implies \sf{m_1 : m_2 = 2 : 7}$

∴ The line segment AB joining (-3, 10) and (6, -8) is divided by (-1, 6) in the ratio 2 : 7

______________________________________