Question

If the point \(P(x,y)\) divides the line segment joining the points \(A(x_1,y_1\)) and \(B(x_2,y_2)\) internally in the ratio \(m_1:m_2\), then \(P(x, y)=\) _____ .

• A. \((\frac {m_1x_2-m_2x_1}{m_1-m_2}, \frac {m_1y_2-m_2y_1}{m_1-m_2})\)
• B. \((\frac {m_1x_2+m_2x_1}{m_1-m_2}, \frac {m_1y_2+m_2y_1}{m_1-m_2})\)
• C. \((\frac {m_1x_2+m_2x_1}{m_1+m_2}, \frac {m_1y_2+m_2y_1}{m_1+m_2})\)
• D. \((\frac {m_1x_2-m_2x_1}{m_1+m_2}, \frac {m_1y_2-m_2y_1}{m_1+m_2})\)

Answer 1

The Correct Option is C

To solve the problem, we need to find the coordinates of point \( P(x, y) \) that divides the line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) internally in the ratio \( m_1 : m_2 \).

1. Section Formula (Internal Division):
If a point divides the line segment joining two points internally in the ratio \( m_1 : m_2 \), then the coordinates of the point are given by:
\[ P(x, y) = \left( \frac{m_1x_2 + m_2x_1}{m_1 + m_2}, \frac{m_1y_2 + m_2y_1}{m_1 + m_2} \right) \]

2. Matching with the Given Options:
The correct option that matches the section formula is:
\[ \left( \frac{m_1x_2 + m_2x_1}{m_1 + m_2}, \frac{m_1y_2 + m_2y_1}{m_1 + m_2} \right) \]

Final Answer:
The correct coordinates of point \( P \) are given in Option (C).