NCERT Class X Chapter 7: Coordinate Geometry Exercise 7.2 Question 8

NCERT Class X Chapter 7: Coordinate Geometry

Question:

If A and B are (–2, –2) and (2, –4), respectively, find the coordinates of P such that AP = (3/7)AB and P lies on the line segment AB.

Given:

  • A = (–2, –2)
  • B = (2, –4)
  • AP = \( \dfrac{3}{7} \) AB
  • P lies on the line segment AB

To Find:

  • Coordinates of P

Formula:

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

\[ P = \left( \frac{mx_2 + nx_1}{m+n},\ \frac{my_2 + ny_1}{m+n} \right) \]

Solution:

Step 1: Let the coordinates of P be \( (x, y) \).

Step 2: Since \( AP = \dfrac{3}{7} AB \), P divides AB in the ratio 3:4 (because AP:PB = 3:4).

Step 3: Using the section formula, substitute the values:

\[ x = \frac{3 \times 2 + 4 \times (-2)}{3 + 4} \] \[ y = \frac{3 \times (-4) + 4 \times (-2)}{3 + 4} \]

Step 4: Simplify the numerators and denominators:

\[ x = \frac{6 + (-8)}{7} = \frac{-2}{7} \] \[ y = \frac{-12 + (-8)}{7} = \frac{-20}{7} \]

Result:

The coordinates of P are \( \left( -\frac{2}{7},\ -\frac{20}{7} \right) \).

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