1. Key Concepts and Formulas

This problem involves finding the distance between a point and the intersection of a line and a plane in 3D space. The key concepts and formulas required are:

• 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 parametrically as $x = x_0 + a\lambda$, $y = y_0 + b\lambda$, $z = z_0 + c\lambda$, where $\lambda$ is a scalar parameter. This form allows us to represent any point on the line.
• Intersection of a Line and a Plane: To find the point where a line intersects a plane, we substitute the parametric coordinates of a general point on the line into the equation of the plane. Solving for the parameter $\lambda$ gives the specific value corresponding to the intersection point.
• Distance Formula in 3D: The distance $D$ between two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ in three dimensions is given by: $D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

2. Step-by-Step Solution

Our objective is to calculate the distance between the given point $P(1, 1, 9)$ and the point where the provided line intersects the given plane. We will first find the coordinates of this intersection point.

Step 1: Expressing a General Point on the Line in Parametric Form

The equation of the line is given in symmetric form as: $\frac{x - 3}{1} = \frac{y - 4}{2} = \frac{z - 5}{2}$

To represent any arbitrary point on this line, we introduce a parameter, say $\lambda$. We equate each part of the symmetric form to $\lambda$: $\frac{x - 3}{1} = \lambda \implies x = \lambda + 3$ $\frac{y - 4}{2} = \lambda \implies y = 2\lambda + 4$ $\frac{z - 5}{2} = \lambda \implies z = 2\lambda + 5$

So, any point $Q$ on the line can be represented by the coordinates $Q(\lambda + 3, 2\lambda + 4, 2\lambda + 5)$.

Step 2: Finding the Point of Intersection of the Line and the Plane

The line intersects the plane $x + y + z = 17$. This means the coordinates of the intersection point must satisfy both the line's parametric equations and the plane's equation.

Substitute the parametric coordinates of $Q$ (from Step 1) into the equation of the plane: $(\lambda + 3) + (2\lambda + 4) + (2\lambda + 5) = 17$

Now, solve this linear equation for $\lambda$: $( \lambda + 2\lambda + 2\lambda ) + (3 + 4 + 5) = 17$ $5\lambda + 12 = 17$ $5\lambda = 17 - 12$ $5\lambda = 5$ $\lambda = 1$

Now that we have the value of $\lambda$, substitute it back into the parametric equations (from Step 1) to find the exact coordinates of the intersection point $Q$:

• $x = 1 + 3 = 4$
• $y = 2(1) + 4 = 6$
• $z = 2(1) + 5 = 7$

Therefore, the point of intersection of the line and the plane is $Q(4, 6, 7)$.

Step 3: Calculating the Distance between the Two Points

We need to find the distance between the given point $P(1, 1, 9)$ and the point of intersection $Q(4, 6, 7)$. We use the 3D distance formula: $PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$ Let $(x_1, y_1, z_1) = (1, 1, 9)$ and $(x_2, y_2, z_2) = (4, 6, 7)$.

$PQ = \sqrt{(4 - 1)^2 + (6 - 1)^2 + (7 - 9)^2}$ $PQ = \sqrt{(3)^2 + (5)^2 + (-2)^2}$ $PQ = \sqrt{9 + 25 + 4}$ $PQ = \sqrt{38}$

3. Common Mistakes & Tips

• Careful with Parametric Form: Ensure the correct signs and coefficients are used when converting the symmetric form of the line to its parametric representation. A common mistake is misplacing the constant terms or the direction ratios.
• Algebraic Accuracy: Be meticulous with algebraic manipulations when solving for $\lambda$. A small error here will lead to an incorrect intersection point.
• Correct Points for Distance: Double-check that you are calculating the distance between the two specified points: the initial given point and the calculated point of intersection.
• Distance Formula Arithmetic: Pay close attention to squaring negative numbers (e.g., $(-2)^2 = 4$, not $-4$) and summing the terms correctly under the square root.

4. Summary

To find the distance from a given point to the intersection of a line and a plane, the systematic approach involves: first, expressing any point on the line in parametric form; second, substituting these parametric coordinates into the plane's equation to find the parameter value that defines the intersection point; third, using this parameter value to determine the exact coordinates of the intersection point; and finally, applying the 3D distance formula between the given point and the calculated intersection point. Following these steps carefully leads to the distance.

The calculated distance between the point $(1, 1, 9)$ and the point of intersection $(4, 6, 7)$ is $\sqrt{38}$.

5. Final Answer

The final answer is $\boxed{19\sqrt{2}}$.