NCERT Class X Chapter 7: Coordinate Geometry Exercise 7.2 Question 5

NCERT Class X Chapter 7: Coordinate Geometry

Question:

Find the ratio in which the line segment joining A(1, –5) and B(–4, 5) is divided by the x-axis. Also find the coordinates of the point of division.

Given:

Points A(1, –5) and B(–4, 5).

To Find:

  • The ratio in which the x-axis divides the line segment AB.
  • The coordinates of the point of division.

Formula:

Section formula: If a point \( P(x, y) \) divides the line segment joining \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \), then

\[ x = \frac{mx_2 + nx_1}{m + n}, \quad y = \frac{my_2 + ny_1}{m + n} \]

Solution:

Step 1: Let the x-axis divide AB at point \( P(x, 0) \) in the ratio \( k:1 \).

Step 2: Using the section formula, the coordinates of \( P \) are:

\[ x = \frac{k \cdot (-4) + 1 \cdot 1}{k + 1} = \frac{-4k + 1}{k + 1} \] \[ y = \frac{k \cdot 5 + 1 \cdot (-5)}{k + 1} = \frac{5k - 5}{k + 1} \]

Step 3: Since the point lies on the x-axis, its y-coordinate is 0.

\[ \frac{5k - 5}{k + 1} = 0 \]

Step 4: Solve for \( k \):

\[ 5k - 5 = 0 \implies 5k = 5 \implies k = 1 \]

Step 5: Therefore, the required ratio is \( 1:1 \).

Step 6: Find the coordinates of the point of division.

\[ x = \frac{-4 \times 1 + 1}{1 + 1} = \frac{-4 + 1}{2} = \frac{-3}{2} = -1.5 \] \[ y = 0 \]

So, the coordinates are \( (-1.5, 0) \).

Result:

The x-axis divides the line segment joining A(1, –5) and B(–4, 5) in the ratio 1:1. The coordinates of the point of division are \( (-1.5, 0) \).

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