The point which divides the line segment joining the points (7, − 6) and (3, − 4) in the ratio 1: 2 internally lies in the :

Here, we are given a line segment with endpoints (7, − 6) and (3, − 4) and we need to find the quadrant in which the coordinates of the point dividing the segment in the ratio 1: 2 lies.

We know that if a point (𝑥, 𝑦) divides the line segment 𝐴(𝑥₁ , 𝑦₁ ) 𝐵(𝑥₂ , 𝑦₂ ) in the ratio 𝑚: 𝑛, then the relation between them is given by

$(x,y)=\left(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n}\right)$

In the given case, (𝑥1 , 𝑦₁ ) = (7, − 6), (𝑥₂ , 𝑦₂ ) = (3, − 4) and 𝑚: 𝑛 = 1: 2 On substituting the values in the relation, we get

$(x,y)=\left(\frac{\left(1\right)\left(3\right)+\left(2\right)\left(7\right)}{1+2},\frac{\left(1\right)\left(4\right)+\left(2\right)(-6)}{1+2}\right)$

$\Rightarrow (x,y)=\left(\frac{17}{3},\frac{-8}{3}\right)$

When a point (𝑥, 𝑦) is defined such that 𝑥 > 0 and 𝑦 < 0, the point lies in 𝐼𝑉^th quadrant.

Hence, the point ($\frac{17}{3}, \frac{-8}{3}$ ) lies in the 𝐼𝑉^th quadrant.