NCERT Class 10 maths exercise 7.4 Question no.1

Class 10 Exercise 7.4 -Q1.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).

Ans.

Let the point of intersection of the given line 2x + y – 4 = 0 and the line segment joining the points A(2,-2) and B(3,7) is (x,y).

Let the point of intersection C(x,y) of line with the line segment AB divides the line segment AB in the ratio of m : n

Applying the section formula

$\boldsymbol{x=\frac{kx_{2}+x_{1}}{k+1}\,, \: y=\frac{ky_{2}+y_{1}}{k+1}}$

Where k = m : n

We are given the A(2,-2) =(x₁,y₁) and B(3,7) = (x₂,y₂)

Putting the value x₁=2, x₂= 3, y₁= -2, y₂=7

$\boldsymbol{x=\frac{3k+2}{k+1}\: ,y=\frac{7k-2}{k+1}}$

Since the point C(x,y) lies in the line 2x + y – 4 = 0, so its co-ordinates (x,y) must satisfy the equation of line.

Therefore putting the value of $\boldsymbol{x=\frac{3k+2}{k+1}\: and\, y=\frac{7k-2}{k+1}}$

In the given equation 2x + y – 4 = 0, we have

$\boldsymbol{2\left ( \frac{3k+2}{k+1} \right )+\frac{7k-2}{k+1}-4=0}$

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

$\boldsymbol{\frac{13k+2}{k+1}=0}$

$\boldsymbol{\frac{9k-2}{k+1}=0}$

$\boldsymbol{9k-2=0}$

$\boldsymbol{k=\frac{2}{9}}$

Therefore the ratio by which the line divides the line segment is 2 : 9

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