Let P(2, -1, 4) and Q (4, 3, 2) are two points and a point R on PQ is…

The correct answer is Option (1) → $\left(\frac{14}{5}, \frac{3}{5}, \frac{16}{5}\right)$

Statement:

Point $R$ lies on $PQ$ such that $3PQ = 5QR$

Reasoning:

From $3PQ = 5QR$, we get:

$\frac{PQ}{QR} = \frac{5}{3}$

So, $QR = \frac{3}{5}PQ$

Hence, $PR = PQ - QR = PQ - \frac{3}{5}PQ = \frac{2}{5}PQ$

Therefore, $R$ divides $PQ$ internally in the ratio:

$PR : RQ = 2 : 3$

Using section formula:

$R = \left( \frac{2x_2 + 3x_1}{5}, \frac{2y_2 + 3y_1}{5}, \frac{2z_2 + 3z_1}{5} \right)$

Substitute $P(2, -1, 4), Q(4, 3, 2)$:

• $x = \frac{2(4) + 3(2)}{5} = \frac{8+6}{5} = \frac{14}{5}$
• $y = \frac{2(3) + 3(-1)}{5} = \frac{6-3}{5} = \frac{3}{5}$
• $z = \frac{2(2) + 3(4)}{5} = \frac{4+12}{5} = \frac{16}{5}$

So, the coordinates of $R$ are $(14/5, 3/5, 16/5)$