Question

A point P divides the line joining points $A(2, 3)$ and $B(10, 7)$ in the ratio 3:1 internally. What are the coordinates of point P?

• A. (6, 6)
• B. (5, 4)
• C. (8, 6)
• D. (4, 5)

Hint:

Use the section formula \(\left(\frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n}\right)\) to find the coordinates of a point dividing a line segment in a given ratio.

Answer 1

The Correct Option is C

The coordinates of point P dividing the line joining points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) in the ratio \(m:n\) are given by the section formula: \[ P\left(\frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n}\right) \] Here, \(A(2, 3)\), \(B(10, 7)\), and the ratio is \(3:1\). Substitute the values into the section formula: \[ P\left(\frac{3 \times 10 + 1 \times 2}{3 + 1}, \frac{3 \times 7 + 1 \times 3}{3 + 1}\right) \] \[ P\left(\frac{30 + 2}{4}, \frac{21 + 3}{4}\right) = P\left(\frac{32}{4}, \frac{24}{4}\right) \] \[ P(8, 6) \]
Final answer
Answer: \(\boxed{(8, 6)}\)