In what ratio, the line joining (-1, 1) and (5, 7) is divided by the…

Question 60 of 70intermediate🔧 ApplyShort Answer4 marks

In what ratio, the line joining $(-1, 1)$ and $(5, 7)$ is divided by the line $x + y = 4$ ?

Correct Answer

The line is divided in the ratio 1:2.

Exercise: Miscellaneous Exercise on Chapter 9 | Q: 13 | (Page 22)

For More Understanding

Explanation

The solution applies the section formula provided in the context (Point II) to find the coordinates of the point dividing the line segment. By assuming the ratio is $k:1$ , we express the coordinates of the intersection point in terms of $k$ . Substituting these coordinates into the equation of the line $x + y = 4$ allows us to solve for $k$ . Since $k$ is positive, the division is internal.

Solution Steps

Let the line segment joining points $A(-1, 1)$ and $B(5, 7)$ be divided by the line $x + y = 4$ at point $P$ in the ratio $k:1$ .
Using the section formula, the coordinates of point $P$ are given by $\left( \frac{k(5) + 1(-1)}{k+1}, \frac{k(7) + 1(1)}{k+1} \right) = \left( \frac{5k - 1}{k+1}, \frac{7k + 1}{k+1} \right)$ .
Since point $P$ lies on the line $x + y = 4$ , it satisfies the equation. Substituting the coordinates of $P$ : $\frac{5k - 1}{k+1} + \frac{7k + 1}{k+1} = 4$ .
Solving the equation:
$\frac{5k - 1 + 7k + 1}{k+1} = 4$ $\frac{12k}{k+1} = 4$ $12k = 4(k+1)$ $12k = 4k + 4$ $8k = 4$
$k = \frac{1}{2}$ .
The required ratio is $k:1$ , which is $\frac{1}{2}:1$ or 1:2.

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