Key Concepts and Formulas

1. Equation of a Line in Symmetric and Parametric Form: A line passing through a point $(x_1, y_1, z_1)$ with direction ratios (D.R.s) $\langle a, b, c \rangle$ can be written as: $\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$ Any point on this line can be represented in parametric form as $(x_1 + a\lambda, y_1 + b\lambda, z_1 + c\lambda)$, where $\lambda$ is a scalar parameter.
2. Condition for a Point to Lie on a Plane: If a point $(x_0, y_0, z_0)$ lies on the plane $Ax + By + Cz + D = 0$, its coordinates must satisfy the plane equation: $Ax_0 + By_0 + Cz_0 + D = 0$.
3. Distance Formula in 3D: The distance between two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ is given by: $PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$ Alternatively, if point $Q$ is $(x_1 + a\lambda, y_1 + b\lambda, z_1 + c\lambda)$, the distance $PQ$ is simply $|\lambda|\sqrt{a^2+b^2+c^2}$.

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Step-by-Step Solution

Step 1: Identify the Direction Ratios of the Line of Measurement

We are given that the distance from point $P(1, -2, 3)$ to the plane $x + 2y - 3z + 10 = 0$ is measured parallel to the line $L$: $\frac{x - 1}{3} = \frac{2 - y}{m} = \frac{z + 3}{1}$ To find the direction ratios (D.R.s) of line $L$, we must express its equation in the standard symmetric form: $\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$. The term $\frac{2 - y}{m}$ can be rewritten as $\frac{-(y - 2)}{m} = \frac{y - 2}{-m}$. So, the standard form of line $L$ is: $\frac{x - 1}{3} = \frac{y - 2}{-m} = \frac{z - (-3)}{1}$ From this, the direction ratios of line $L$ are $\langle 3, -m, 1 \rangle$. Since the distance is measured parallel to line $L$, the line segment $PQ$ (where $Q$ is on the plane) will have these same direction ratios.

Step 2: Formulate the Parametric Equation of the Line Segment PQ

The line segment $PQ$ passes through the point $P(1, -2, 3)$ and has direction ratios $\langle 3, -m, 1 \rangle$. The parametric equation of this line can be written as: $\frac{x - 1}{3} = \frac{y - (-2)}{-m} = \frac{z - 3}{1} = \lambda$ where $\lambda$ is a scalar parameter. Any point $Q$ on this line can be represented by its coordinates in terms of $\lambda$: $Q = (1 + 3\lambda, -2 - m\lambda, 3 + \lambda)$

Step 3: Find the Value of Parameter $\lambda$ for Point Q

Point $Q$ lies on the plane $x + 2y - 3z + 10 = 0$. Therefore, its coordinates must satisfy the plane equation. Substitute the parametric coordinates of $Q$ into the plane equation: $(1 + 3\lambda) + 2(-2 - m\lambda) - 3(3 + \lambda) + 10 = 0$ Now, simplify and solve for $\lambda$: $1 + 3\lambda - 4 - 2m\lambda - 9 - 3\lambda + 10 = 0$ Group terms with $\lambda$ and constant terms: $(3\lambda - 2m\lambda - 3\lambda) + (1 - 4 - 9 + 10) = 0$ $-2m\lambda - 2 = 0$ $-2m\lambda = 2$ $m\lambda = -1$ From this, we find the value of $\lambda$: $\lambda = -\frac{1}{m}$

Step 4: Calculate the Distance PQ

The distance $PQ$ between $P(1, -2, 3)$ and $Q(1 + 3\lambda, -2 - m\lambda, 3 + \lambda)$ can be calculated using the distance formula for points on a line: $PQ = |\lambda| \sqrt{(\text{D.R.}_x)^2 + (\text{D.R.}_y)^2 + (\text{D.R.}_z)^2}$ $PQ = |\lambda| \sqrt{3^2 + (-m)^2 + 1^2}$ $PQ = |\lambda| \sqrt{9 + m^2 + 1}$ $PQ = |\lambda| \sqrt{10 + m^2}$ Substitute the value of $\lambda = -\frac{1}{m}$: $PQ = \left|-\frac{1}{m}\right| \sqrt{10 + m^2}$ $PQ = \frac{1}{|m|} \sqrt{10 + m^2}$

Step 5: Equate the Calculated Distance to the Given Distance and Solve for |m|

We are given that the distance $PQ = \sqrt{\frac{7}{2}}$. Set up the equation: $\frac{1}{|m|} \sqrt{10 + m^2} = \sqrt{\frac{7}{2}}$ Square both sides of the equation to eliminate the square roots: $\left(\frac{1}{|m|}\right)^2 (10 + m^2) = \frac{7}{2}$ $\frac{1}{m^2} (10 + m^2) = \frac{7}{2}$ Distribute $\frac{1}{m^2}$: $\frac{10}{m^2} + \frac{m^2}{m^2} = \frac{7}{2}$ $\frac{10}{m^2} + 1 = \frac{7}{2}$ Subtract 1 from both sides: $\frac{10}{m^2} = \frac{7}{2} - 1$ $\frac{10}{m^2} = \frac{7 - 2}{2}$ $\frac{10}{m^2} = \frac{5}{2}$ Now, solve for $m^2$: $5m^2 = 10 \times 2$ $5m^2 = 20$ $m^2 = \frac{20}{5}$ $m^2 = 4$ Finally, take the square root to find $|m|$: $|m| = \sqrt{4}$ $|m| = 2$

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Common Mistakes & Tips

• Incorrect Direction Ratios: Always ensure the line equation is in the standard symmetric form ($\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}$) before identifying direction ratios. Pay close attention to terms like $(2-y)$.
• Algebraic Errors: Be careful with substitutions and simplification, especially when dealing with negative signs and fractions involving the parameter $\lambda$ and $m$.
• Absolute Value: Remember that distance is always positive, so use $|\lambda|$ and $|m|$ when appropriate in the distance calculation.

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Summary

To find the value of $|m|$, we first determined the direction ratios of the line along which the distance is measured. Then, we used these direction ratios and the given point to write the parametric equation of the line segment connecting the point to the plane. By substituting the parametric coordinates into the plane equation, we found the value of the parameter $\lambda$. Finally, we used the distance formula, incorporating the value of $\lambda$, and equated it to the given distance to solve for $|m|$. The calculation consistently leads to $|m|=2$.

The final answer is $\boxed{2}$.