Question 13 of 18intermediate🔧 ApplyNumerical3 marks

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.

Correct Answer

The line segment joining A(1, -5) and B(-4, 5) is divided by the x-axis in the ratio 1:1. The coordinates of the point of division are (-3/2, 0).

Exercise: EXERCISE 7.2 | Q: 5 | (Page 13)
For More Understanding

Explanation

This question is similar to Example 9 in the textbook, which finds the ratio when the y-axis divides a line segment. Here, we use the same approach but for the x-axis. The key insight is that any point on the x-axis has its y-coordinate equal to 0. Using the section formula with ratio k:1, we equate the y-coordinate to 0 to find k, then substitute back to find the x-coordinate.

Solution Steps

  1. Step 1: Let the point on the x-axis that divides the line segment joining A(1, -5) and B(-4, 5) be P(x, 0). Since P lies on the x-axis, its y-coordinate is 0.

  2. Step 2: Let the ratio be k:1. Using the section formula, the coordinates of P are: P = ((k(-4) + 1(1))/(k+1), (k(5) + 1(-5))/(k+1)) = ((-4k + 1)/(k+1), (5k - 5)/(k+1))

  3. Step 3: Since P lies on the x-axis, y-coordinate = 0. Therefore: (5k - 5)/(k+1) = 0, which gives 5k - 5 = 0, so k = 1.

  4. Step 4: Therefore, the ratio is 1:1.

  5. Step 5: Finding the coordinates of P: x = (-4k + 1)/(k+1) = (-4(1) + 1)/(1+1) = -3/2, and y = 0. Therefore, the coordinates of the point of division are (-3/2, 0).

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