NCERT Class X Chapter 7: Coordinate Geometry Example 6
NCERT Class X Chapter 7: Coordinate Geometry Example 6
Question:
Find the coordinates of the point which divides the line segment joining the points (4, –3) and (8, 5) in the ratio 3 : 1 internally.
Given:
- Point A: (4, –3)
- Point B: (8, 5)
- The point divides AB in the ratio 3:1 internally.
To Find:
The coordinates of the point that divides the line segment AB in the ratio 3:1 internally.
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 \( m:n \) internally, then:
$$ x = \frac{mx_2 + nx_1}{m + n}, \qquad y = \frac{my_2 + ny_1}{m + n} $$Solution:
Step 1: Assign values to the variables.
Let \( A(x_1, y_1) = (4, -3) \), \( B(x_2, y_2) = (8, 5) \), and the ratio \( m:n = 3:1 \).
Step 2: Write the section formula for \( x \)-coordinate.
$$ x = \frac{m x_2 + n x_1}{m + n} $$Substitute the values:
$$ x = \frac{3 \times 8 + 1 \times 4}{3 + 1} $$Step 3: Calculate the value of \( x \).
$$ x = \frac{24 + 4}{4} = \frac{28}{4} = 7 $$Step 4: Write the section formula for \( y \)-coordinate.
$$ y = \frac{m y_2 + n y_1}{m + n} $$Substitute the values:
$$ y = \frac{3 \times 5 + 1 \times (-3)}{3 + 1} $$Step 5: Calculate the value of \( y \).
$$ y = \frac{15 - 3}{4} = \frac{12}{4} = 3 $$Step 6: Write the coordinates of the required point.
$$ \text{Coordinates} = (7, 3) $$Result:
Therefore, the coordinates of the point that divides the line segment joining (4, –3) and (8, 5) in the ratio 3:1 internally are (7, 3).
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