Determine the ratio in which the line 2x+y−4=0 divides the line segment joining the points A(2,−2) and B(3,7).

Question

Determine the ratio in which the line 2x+y−4=0 divides the line segment joining the points A(2,−2) and B(3,7).

Filo · Accepted Answer

Let the given line divides the line segment joining the points

A (2, - 2) \equiv (x_{1}, y_{1})

and

B (3, 7) \equiv (x_{2}, y_{2})

in ratio

m_{1} : m_{2} = k : 1

∴ x = \frac{m _{1} x _{2} + m _{2} x _{1}}{m _{1} + m _{2}}

\Rightarrow x = \frac{k \times 3 + 1 \times 2}{k + 1} = \frac{3 k + 2}{k + 1}

And

y = \frac{m _{1} y _{2} + m _{2} y _{1}}{m _{1} + m _{2}}

= \frac{k \times 7 + 1 \times - 2}{k + 1}

= \frac{7 k - 2}{k + 1}

This point also lies on

2 x + y - 4 = 0

Put the value of x and y in the given equation

∴ 2 \times [\frac{3 k + 2}{k + 1}] + [\frac{7 k - 2}{k + 1}] - 4 = 0

= \frac{6 k + 4 + 7 k - 2 - 4 k - 4}{k + 1} = 0

\Rightarrow 9 k - 2 = 0

\Rightarrow k = \frac{2}{9}

∴

The required ratio is

2 : 9.

Question Text	Determine the ratio in which the line $2 x + y - 4 = 0$ divides the line segment joining the points $A (2, - 2)$ and $B (3, 7)$ .
Answer Type	Text solution:1
Upvotes	150

Determine the ratio in which the line $2 x + y - 4 = 0$ divides the line segment joining the points $A (2, - 2)$ and $B (3, 7)$ .