Key Concepts and Formulas

• Image of a Point in a Plane Measured Parallel to a Line: If Q is the image of point P in a plane $\Pi$ measured parallel to a line L, then the line segment PQ is parallel to line L. Crucially, the point of intersection (let's call it F) of the line PQ with the plane $\Pi$ is the midpoint of the segment PQ.
• Equation of a Line: A line passing through a point $(x_1, y_1, z_1)$ with direction ratios $(a, b, c)$ can be represented in symmetric form as $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$. Its parametric form is $(x_1 + a\lambda, y_1 + b\lambda, z_1 + c\lambda)$.
• Distance Formula in 3D: The distance between two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is given by $\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$.

Step-by-Step Solution

Step 1: Determine the Equation of the Line PQ

• Understanding the problem statement: We are given point P(1, -2, 3) and a plane $2x + 3y - 4z + 22 = 0$. The image Q is measured parallel to the line $\frac{x}{1} = \frac{y}{4} = \frac{z}{5}$. This means the line segment PQ is parallel to the given line.
• Identifying direction ratios: The direction ratios of the given line are (1, 4, 5). Since line PQ is parallel to this line, the direction ratios of line PQ are also (1, 4, 5).
• Formulating the line equation: The line PQ passes through point P(1, -2, 3) and has direction ratios (1, 4, 5). Therefore, its equation in symmetric form is: $\frac{x - 1}{1} = \frac{y - (-2)}{4} = \frac{z - 3}{5}$ $\frac{x - 1}{1} = \frac{y + 2}{4} = \frac{z - 3}{5}$
• Parametric representation: To find the point of intersection, we express any general point F on the line PQ in parametric form by setting each part of the equation equal to a parameter $\lambda$: $x = \lambda + 1$ $y = 4\lambda - 2$ $z = 5\lambda + 3$ So, any point F on the line PQ has coordinates $(\lambda + 1, 4\lambda - 2, 5\lambda + 3)$.

Step 2: Find the Coordinates of the Point of Intersection F

• Condition for intersection: The point F, where the line PQ intersects the plane $2x + 3y - 4z + 22 = 0$, must satisfy the equation of the plane.
• Substitute F into the plane equation: Substitute the parametric coordinates of F into the plane's equation: $2(\lambda + 1) + 3(4\lambda - 2) - 4(5\lambda + 3) + 22 = 0$
• Solve for $\lambda$: $2\lambda + 2 + 12\lambda - 6 - 20\lambda - 12 + 22 = 0$ Combine the $\lambda$ terms and the constant terms: $(2 + 12 - 20)\lambda + (2 - 6 - 12 + 22) = 0$ $-6\lambda + 6 = 0$ $-6\lambda = -6$ $\lambda = 1$
• Determine the coordinates of F: Substitute the value of $\lambda = 1$ back into the parametric equations for F: $F_x = 1 + 1 = 2$ $F_y = 4(1) - 2 = 2$ $F_z = 5(1) + 3 = 8$ Thus, the coordinates of the point of intersection F are (2, 2, 8).

Step 3: Calculate the Distance PQ

• Applying the midpoint property: As per the key concept, the point F (2, 2, 8) is the midpoint of the segment PQ. This implies that the distance from P to F is equal to the distance from F to Q, and the total distance PQ is twice the distance PF. $PQ = 2 \times PF$
• Calculate PF using the distance formula: We have P(1, -2, 3) and F(2, 2, 8). $PF = \sqrt{(2 - 1)^2 + (2 - (-2))^2 + (8 - 3)^2}$ $PF = \sqrt{(1)^2 + (4)^2 + (5)^2}$ $PF = \sqrt{1 + 16 + 25}$ $PF = \sqrt{42}$
• Calculate PQ: $PQ = 2 \times PF = 2 \times \sqrt{42}$ $PQ = 2\sqrt{42}$

Common Mistakes & Tips

• Distinguish Image Types: Always differentiate between "image measured parallel to a line" and "reflection" (or "image in a plane"). For reflection, the line PQ would be perpendicular to the plane, and its direction ratios would be proportional to the normal vector of the plane.
• Midpoint Property is Key: Remember that the point of intersection (F) of the line PQ with the plane is always the midpoint of the segment connecting the original point P and its image Q. This relationship is crucial for calculating the distance PQ efficiently.
• Parametric Form Efficiency: Using the parametric form for points on a line is a standard and highly efficient technique for finding intersections in 3D geometry problems.

Summary

To find the distance PQ, where Q is the image of P in a plane measured parallel to a given line, we first determine the equation of the line passing through P and parallel to the given line. Then, we find the point of intersection, F, of this line with the plane. This point F is the midpoint of the segment PQ. Finally, we calculate the distance PF using the distance formula and double it to get the required distance PQ. Following these steps, we found the distance PQ to be $2\sqrt{42}$.

The final answer is \boxed{\text{2\sqrt {42}}}, which corresponds to option (A).