Key Concepts and Formulas

To solve this problem, we'll utilize fundamental principles from 3D Coordinate Geometry:

1. Equation of a Line in 3D (Symmetric Form): When a line passes through two distinct points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$, its equation can be expressed in symmetric form as: $\frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1}$ This formula is essential because point $R$ lies on the line segment $PQ$, meaning its coordinates must satisfy this equation. We will use this to find the unknown $y$ and $z$ coordinates of $R$.

2. Distance Formula in 3D: The distance between any two points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ in 3D space is given by: $D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$ For the specific case of finding the distance of a point $R(x, y, z)$ from the origin $O(0, 0, 0)$, the formula simplifies to: $RO = \sqrt{x^2 + y^2 + z^2}$ Once we determine the complete coordinates of point $R$, we will use this simplified formula to calculate its distance from the origin.

Step-by-Step Solution

Step 1: Determine the Equation of the Line Passing Through P and Q

• Objective: To find the algebraic relationship that defines all points lying on the line segment $PQ$. Since point $R$ lies on this segment, its coordinates must satisfy this equation.
• Given Points: We are given $P(x_1, y_1, z_1) = (2, -3, 4)$ and $Q(x_2, y_2, z_2) = (8, 0, 10)$.

We substitute the coordinates of $P$ and $Q$ into the symmetric form of the line equation: $\frac{x - 2}{8 - 2} = \frac{y - (-3)}{0 - (-3)} = \frac{z - 4}{10 - 4}$

Now, we simplify the denominators to obtain the equation of the line: $\frac{x - 2}{6} = \frac{y + 3}{3} = \frac{z - 4}{6}$ This equation represents all points on the line extending through $P$ and $Q$.

Step 2: Find the Unknown Coordinates (y and z) of Point R

• Objective: To fully determine the coordinates of point $R$. We are given $R(4, y, z)$ and know it lies on the line segment $PQ$.
• Reasoning: Since point $R$ lies on the line $PQ$, its coordinates $(4, y, z)$ must satisfy the line equation derived in Step 1.

Substitute the known $x$-coordinate of $R$, which is $x=4$, into the line equation: $\frac{4 - 2}{6} = \frac{y + 3}{3} = \frac{z - 4}{6}$

Simplify the first term: $\frac{2}{6} = \frac{y + 3}{3} = \frac{z - 4}{6}$ $\frac{1}{3} = \frac{y + 3}{3} = \frac{z - 4}{6}$

Now, we equate the ratios to solve for $y$ and $z$ independently:

• To find the y-coordinate: Equate the first and second parts of the equation: $\frac{1}{3} = \frac{y + 3}{3}$ Multiply both sides by 3: $1 = y + 3$ Subtract 3 from both sides: $y = 1 - 3$ $y = -2$

• To find the z-coordinate: Equate the first and third parts of the equation: $\frac{1}{3} = \frac{z - 4}{6}$ Multiply both sides by 6: $6 \times \frac{1}{3} = z - 4$ $2 = z - 4$ Add 4 to both sides: $z = 2 + 4$ $z = 6$

Thus, the complete coordinates of point $R$ are $(4, -2, 6)$.

• Verification for "Line Segment": The problem specifies that $R$ lies on the line segment $PQ$. We can verify this by checking if the coordinates of $R$ lie between the corresponding coordinates of $P$ and $Q$:
  • For $x$: $P_x=2$, $R_x=4$, $Q_x=8$. ($2 \le 4 \le 8$) - True.
  • For $y$: $P_y=-3$, $R_y=-2$, $Q_y=0$. ($-3 \le -2 \le 0$) - True.
  • For $z$: $P_z=4$, $R_z=6$, $Q_z=10$. ($4 \le 6 \le 10$) - True. Since all conditions are met, $R$ indeed lies on the line segment $PQ$.

Step 3: Calculate the Distance of R from the Origin

• Objective: To find the final answer, which is the distance of point $R$ from the origin $O(0,0,0)$.
• Given Points: We have $R(x, y, z) = (4, -2, 6)$ and the Origin $O(0, 0, 0)$.
• Reasoning: We use the simplified 3D distance formula for a point from the origin.

Substitute the coordinates of $R$ into the distance formula $RO = \sqrt{x^2 + y^2 + z^2}$: $RO = \sqrt{(4)^2 + (-2)^2 + (6)^2}$ $RO = \sqrt{16 + 4 + 36}$ $RO = \sqrt{56}$

To simplify the radical, we look for perfect square factors of 56. We know that $56 = 4 \times 14$: $RO = \sqrt{4 \times 14}$ $RO = \sqrt{4} \times \sqrt{14}$ $RO = 2 \sqrt{14}$

Therefore, the distance of point $R$ from the origin is $2 \sqrt{14}$.

Common Mistakes & Tips

• Sign Errors: Be extremely careful with negative signs when substituting coordinates into the line equation and especially in the distance formula (e.g., $(-2)^2$ must be $4$, not $-4$).
• Simplifying Radicals: Always simplify square roots to their simplest form, as shown in Step 3. This often involves factoring out perfect squares.
• "Line" vs. "Line Segment": While the line equation finds points on the infinite line, the term "line segment" implies the point must lie between the given two points. A quick check of coordinates (as done in Step 2) confirms this. Alternatively, one could use the section formula: if $R$ divides $PQ$ in ratio $\lambda:1$, for $R$ to be on the segment, $\lambda$ must be positive.

Summary

This problem required us to first determine the full coordinates of point $R$ by utilizing the fact that it lies on the line segment connecting points $P$ and $Q$. We achieved this by finding the symmetric equation of the line $PQ$ and substituting the known $x$-coordinate of $R$ to solve for its $y$ and $z$ coordinates. Once $R(4, -2, 6)$ was fully determined, we applied the 3D distance formula to calculate its distance from the origin. The final distance was found to be $2\sqrt{14}$.

The final answer is $\boxed{2 \sqrt {14}}$, which corresponds to option (A).