Question

The point P divides the line segment joining the points (1, 2) and (−1, 2) internally in the ratio 1 : 2. 

Assertion (A): The co-ordinates of point P = (1, 6). 

Reason (R): If point P divides line segment joining the points \[(x_1, y_1)\] and \[(x_2, y_2)\] in the ratio \[m_1:m_2\] then \[\mathrm{P}=\left(\frac{m_1 x_2+m_2 x_1}{m_1+m_2}, \frac{m_1 y_2+m_2 y_1}{m_1+m_2}\right)\]

विकल्प

• A is true, R is false.

• A is false, R is true.

• Both A and R are true and R is the correct reason for A.

• Both A and R are true and R is the incorrect reason for A.

Answer 1

A is false, R is true.

Explanation:

We know that,

If point P divides the line segment joining the points \[(x_1, y_1)\] and \[(x_2, y_2)\] in the ratio \[m_1 : m_2\], then:

Co-ordinates of \[P = \left(\frac{m_1x_2 + m_2x_1}{m_1 + m_2}, \frac{m_1y_2 + m_2y_1}{m_1 + m_2}\right)\]

So, reason (R) is true.

Here, \[(x_1, y_1) = (1, 2)\] and \[(x_2, y_2) = (-1, 2)\]

\[m_1 : m_2 = 1 : 2\]

Substituting the values, we get:

⇒ \[P = \left[\frac{1 \times (-1) + 2 \times 1}{1 + 2}, \frac{1 \times 2 + 2 \times 2}{1 + 2}\right]\]

\[{} = \left(\frac{-1 + 2}{3}, \frac{2 + 4}{3}\right)\]

\[{} = \left(\frac{1}{3}, \frac{6}{3}\right)\]

\[{} = \left(\frac{1}{3}, 2\right)\]

So, assertion (A) is false.