Question

Find in what ratio line x+y=4 divide the line joining the points (-1,1)and (5,7)

Answer 1

Answer:

Let the ratio be k:1

now

coordinates of the point of intersection :

5k-1/k+1 , 7k+1/k+1

now

this point must satisfy the equation of the line

therefore

5k-1/k+1 + 7k+1/k+1 = 4

12k/k+1 = 4

12k = 4k+4

8k = 4

K = 4/8 = 1/2

therefore

the line divide it in the ratio 1:2

Answer 2

Answer:

Ratio of division = 1 : 2

Step-by-step explanation:

Let's assume the point of Intersection be (x, y) and also assume that the line x+y=4 divides the line joining the given points in the ratio λ : 1.

Now, we are using section formula to find the ratio of division.

Section formula:-

• $\sf(x,y) = \left( \dfrac{m_1x_2 + m_2x_1}{m_1 + m_2}, \dfrac{m_1y_2 + m_2y_1}{m_1 + m_2} \right)$

Here, m1 : m2 = λ : 1.

By substituting the values in the formula, we get:

$\sf \implies(x,y) = \left( \dfrac{5 \lambda + ( - 1)}{ \lambda + 1}, \dfrac{7 \lambda + 1}{ \lambda + 1} \right)$

$\sf \implies(x,y) = \left( \dfrac{5 \lambda - 1}{ \lambda + 1}, \dfrac{7 \lambda + 1}{ \lambda + 1} \right)$

Now, since these coordinates are the point of Intersection, they must satisfy the given equation of line.

$\sf \implies x + y = 4$

$\sf \implies \dfrac{5 \lambda - 1}{ \lambda + 1} + \dfrac{7 \lambda + 1}{ \lambda + 1} = 4$

$\sf \implies \dfrac{5 \lambda - 1 + 7 \lambda + 1}{ \lambda + 1}= 4$

$\sf \implies \dfrac{12 \lambda }{ \lambda + 1}= 4$

$\sf \implies 3 \lambda = \lambda + 1$

$\sf \implies 2 \lambda = 1$

$\boxed{ \sf \implies \lambda = \dfrac{1}{2} }$

Therefore the ratio of division = λ : 1 = 1/2 : 1 or 1:2