Key Concepts and Formulas

• Area of a Triangle: The area of a triangle can be calculated using the formula $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$. In 3D geometry, the height is the perpendicular distance from a vertex to the line containing its opposite side (the base).
• Parametric Form of a Line: A line given in symmetric form $\frac{x-x_0}{a}=\frac{y-y_0}{b}=\frac{z-z_0}{c}$ can be expressed in parametric form as $x = x_0 + a\lambda$, $y = y_0 + b\lambda$, $z = z_0 + c\lambda$, where $(x_0, y_0, z_0)$ is a point on the line and $(a,b,c)$ are its direction ratios.
• Perpendicularity of Vectors: Two vectors $\vec{u}$ and $\vec{v}$ are perpendicular if and only if their dot product is zero, i.e., $\vec{u} \cdot \vec{v} = 0$.

Step-by-Step Solution

Step 1: Representing a General Point on the Line QR

• What and Why: We begin by expressing any general point on the line containing the base QR in terms of a single parameter. This allows us to represent the foot of the perpendicular (let's call it H) from P to the line QR, which is essential for finding the height.
• Math: The equation of the line on which Q and R lie is given as: $\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}$ Let's set this equal to a parameter $\lambda$: $\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3} = \lambda$ From this, we can write the coordinates of any point H on the line as: $H(5\lambda - 3, 2\lambda + 1, 3\lambda - 4)$
• Explanation: This parametric representation is a fundamental tool in 3D geometry, allowing us to refer to any point on the line using a single variable, $\lambda$.

Step 2: Defining the Vector $\vec{PH}$

• What and Why: We need to find the height of the triangle, which is the perpendicular distance from point P to the line QR. To do this, we first form a vector from the point P to the general point H on the line.
• Math: The coordinates of point P are $(0,2,3)$. The coordinates of the general point H are $(5\lambda - 3, 2\lambda + 1, 3\lambda - 4)$. The vector $\vec{PH}$ is obtained by subtracting the coordinates of P from H: $\vec{PH} = ( (5\lambda - 3) - 0, (2\lambda + 1) - 2, (3\lambda - 4) - 3 )$ $\vec{PH} = (5\lambda - 3, 2\lambda - 1, 3\lambda - 7)$
• Explanation: This vector represents the segment connecting vertex P to any point on the line QR. When H is the foot of the perpendicular, the magnitude of this vector will be the height.

Step 3: Using the Perpendicularity Condition

• What and Why: The height of the triangle is the perpendicular distance from P to the line QR. This means the vector $\vec{PH}$ (where H is the foot of the perpendicular) must be perpendicular to the direction vector of the line QR. We use the dot product property for perpendicular vectors.
• Math: The direction ratios of the line QR are given by the denominators in its symmetric equation, which are $(5,2,3)$. So, the direction vector of the line QR is $\vec{d} = (5,2,3)$. Since $\vec{PH}$ is perpendicular to the line QR, their dot product must be zero: $\vec{PH} \cdot \vec{d} = 0$ $(5\lambda - 3)(5) + (2\lambda - 1)(2) + (3\lambda - 7)(3) = 0$
• Explanation: This step translates the geometric condition of perpendicularity into an algebraic equation involving $\lambda$, which we can then solve.

Step 4: Solving for $\lambda$

• What and Why: We solve the equation obtained from the dot product to find the specific value of $\lambda$. This value of $\lambda$ will correspond to the unique point H on the line that is closest to P (i.e., the foot of the perpendicular).
• Math: Expanding and simplifying the equation: $25\lambda - 15 + 4\lambda - 2 + 9\lambda - 21 = 0$ Combine the terms with $\lambda$ and the constant terms: $(25\lambda + 4\lambda + 9\lambda) + (-15 - 2 - 21) = 0$ $38\lambda - 38 = 0$ $38\lambda = 38$ $\lambda = 1$
• Explanation: The solution for $\lambda$ pinpoints the exact location of the foot of the perpendicular H on the line QR.

Step 5: Finding the Coordinates of H

• What and Why: Now that we have the value of $\lambda$, we substitute it back into the parametric equations for H to find its exact coordinates. H is the foot of the perpendicular from P to the line QR.
• Math: Substitute $\lambda = 1$ into $H(5\lambda - 3, 2\lambda + 1, 3\lambda - 4)$:
  • $x_H = 5(1) - 3 = 2$
  • $y_H = 2(1) + 1 = 3$
  • $z_H = 3(1) - 4 = -1$ So, the coordinates of H are $(2,3,-1)$.
• Explanation: With the specific coordinates of H, we can now calculate the length of the segment PH, which is the height of the triangle.

Step 6: Calculating the Height PH

• What and Why: The height of the triangle is the distance between point P and the foot of the perpendicular H. We use the 3D distance formula to calculate this length.
• Math: The coordinates of P are $(0,2,3)$ and H are $(2,3,-1)$. Using the distance formula $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$: $PH = \sqrt{(2 - 0)^2 + (3 - 2)^2 + (-1 - 3)^2}$ $PH = \sqrt{(2)^2 + (1)^2 + (-4)^2}$ $PH = \sqrt{4 + 1 + 16}$ $PH = \sqrt{21}$
• Explanation: This value, $\sqrt{21}$, is the height of the triangle PQR.

Step 7: Calculating the Area of Triangle PQR

• What and Why: We now have both the base length QR and the height PH. We can directly apply the area formula for a triangle.
• Math: We are given that the length of the base QR is 5. We calculated the height $PH = \sqrt{21}$. Using the area formula: $\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$ $\text{Area} = \frac{1}{2} \times QR \times PH$ $\text{Area} = \frac{1}{2} \times 5 \times \sqrt{21}$ $\text{Area} = \frac{5\sqrt{21}}{2}$
• Explanation: All necessary components are in place to compute the area of the triangle.

Step 8: Comparing with the Given Form $\frac{m}{n}$ and Checking Options

• What and Why: The problem states the area is $\frac{m}{n}$. We need to compare our calculated area with this form and then check which of the given options holds true.
• Math: Our calculated area is $\frac{5\sqrt{21}}{2}$. So, we can set $\frac{m}{n} = \frac{5\sqrt{21}}{2}$. This suggests $m = 5\sqrt{21}$ and $n=2$ (or scalar multiples thereof). Let's check Option (A): $2m - 5\sqrt{21}n = 0$. Substitute $m = 5\sqrt{21}$ and $n=2$: $2(5\sqrt{21}) - 5\sqrt{21}(2) = 10\sqrt{21} - 10\sqrt{21} = 0$ Since the equation holds true, Option (A) is the correct choice.
• Explanation: This final step confirms that our calculated area matches one of the given relationships between $m$ and $n$.

Common Mistakes & Tips

• Parametric Form Errors: Ensure correct algebraic manipulation when converting the symmetric form of a line to its parametric form. A common error is sign mistakes or incorrect coefficient association.
• Vector Direction: Always use the direction vector of the line for the perpendicularity condition, not just any point on the line. The direction vector is derived from the denominators of the symmetric equation.
• Calculation Accuracy: Be meticulous with arithmetic, especially when dealing with squares and square roots in the distance formula and during the dot product calculation.

Summary

This problem effectively tests the application of 3D geometry concepts to find the area of a triangle. The core idea is to determine the height of the triangle by finding the perpendicular distance from a vertex to the line containing its opposite side. This involves representing a general point on the line using a parametric form, forming a vector from the vertex to this general point, and then using the perpendicularity condition (dot product equals zero) to locate the foot of the perpendicular. Once the height is found, the standard area formula is applied. The problem concludes by matching the calculated area with the given options.

The final answer is $\boxed{\text{A}}$.