The point P divides the line segment joining the point (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 the line segment joining the points (x₁, y₁) and (x₂, y₂) in the ratio m₁ : m₂ then :

P = $\Big(\dfrac{m$1x2 + m2x1}{m1 + m2}, \dfrac{m1y2 + m2y1}{m1 + m2}\Big)

1. A is true, R is false.

2. A is false, R is true.

3. Both A and R are true and R is correct reason for A.

4. Both A and R are true and R is incorrect reason for A.

We know that,

If point P divides the line segment joining the points (x₁, y₁) and (x₂, y₂) in the ratio m₁ : m₂, then :

Co-ordinates of P = $\Big(\dfrac{m$1x2 + m2x1}{m1 + m2}, \dfrac{m1y2 + m2y1}{m1 + m2}\Big)

So, reason (R) is true.

Here, (x₁, y₁) = (1, 2) and (x₂, y₂) = (-1, 2)

m₁ : m₂ = 1 : 2

Substituting the values, we get :

$\Rightarrow \text{P} = \Big[\dfrac{1\times (-1) + 2\times 1}{1 + 2}, \dfrac{1 \times 2 + 2 \times 2}{1 + 2}\Big]\\[1em] = \Big(\dfrac{-1 + 2}{3}, \dfrac{2 + 4}{3}\Big)\\[1em] = \Big(\dfrac{1}{3}, \dfrac{6}{3}\Big)\\[1em] = \Big(\dfrac{1}{3}, 2\Big)\\[1em]$

So, assertion (A) is false.

Hence, option 2 is the correct option.