This solution will guide you through solving the problem by leveraging key geometric properties of a parabola, rather than resorting to complex coordinate geometry equations. We will focus on the relationship between the focus, the tangent at the vertex, and the latus rectum.

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Key Concepts and Formulas

To solve this problem efficiently, we rely on two fundamental properties of a parabola:

1. Focal Length Definition: For any parabola, the focal length, often denoted by $p$, is defined as the perpendicular distance from the focus ($F$) to the tangent at its vertex ($T_V$). $p = \text{Perpendicular distance}(F, T_V)$ It's crucial to remember that focal length $p$ is always a positive value, as it represents a distance.

2. Latus Rectum and Focal Length Relationship: The length of the latus rectum of a parabola is always four times its focal length. $\text{Length of Latus Rectum} = 4p$

Additionally, we will use the formula for the perpendicular distance from a point $(x_1, y_1)$ to a line $Ax+By+C=0$: $d = \frac{|Ax_1+By_1+C|}{\sqrt{A^2+B^2}}$

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Step-by-Step Derivation

Step 1: Understanding the Problem and Identifying Given Information

Our objective is to find the value of $|a|$. We are provided with the following specific details about a parabola:

• Focus of the parabola: $F = (a, a)$. This is the fixed point that defines the parabola.
• Equation of the tangent at its vertex: $T_V: x + y = a$. This is a crucial line, as its perpendicular distance from the focus gives us the focal length. To use the perpendicular distance formula, we rewrite this in the standard form $Ax+By+C=0$: $x + y - a = 0$. Here, $A=1$, $B=1$, and $C=-a$.
• Length of the latus rectum: 16. This provides a numerical value that will allow us to determine the focal length and subsequently $|a|$.

Step 2: Calculating the Focal Length ($p$) of the Parabola

• Why this step? The focal length $p$ is the bridge connecting the given geometric elements (focus and tangent at vertex) to the latus rectum. By calculating $p$ first, we can then use the given latus rectum length to solve for the unknown parameter $a$. The key concept states that the perpendicular distance from the focus to the tangent at the vertex is the focal length.

• We will use the perpendicular distance formula with the point $(x_1, y_1) = (a, a)$ (the focus) and the line $x+y-a=0$ (the tangent at the vertex). Substituting the values into the formula: $p = \frac{|1(a) + 1(a) - a|}{\sqrt{1^2 + 1^2}}$ $p = \frac{|a + a - a|}{\sqrt{1 + 1}}$ $p = \frac{|a|}{\sqrt{2}}$

• Important Note: As focal length $p$ represents a physical distance, it must always be a non-negative value. The absolute value $|a|$ in the numerator correctly ensures that $p$ is positive, regardless of the sign of $a$.

Step 3: Utilizing the Latus Rectum Information

• Why this step? We have an expression for the focal length $p$ in terms of $a$. The problem also provides the numerical value of the latus rectum. By using the fundamental relationship between the latus rectum and the focal length, we can form an equation that allows us to solve for $a$.

• From the key concept, we know that the length of the latus rectum is $4p$.

• Now, we substitute the expression for $p$ that we found in Step 2 into this relationship: $\text{Length of Latus Rectum} = 4 \times \left(\frac{|a|}{\sqrt{2}}\right)$

Step 4: Solving for the Unknown Parameter $|a|$

• Why this step? We now have a direct equation relating the given length of the latus rectum (16) to the unknown $|a|$. This is the final algebraic step to isolate and find the value of $|a|$.

• We are given that the length of the latus rectum is 16. Therefore, we can set up the equation: $4 \times \frac{|a|}{\sqrt{2}} = 16$

• Now, we solve this equation for $|a|$:

  1. Divide both sides of the equation by 4: $\frac{|a|}{\sqrt{2}} = \frac{16}{4}$ $\frac{|a|}{\sqrt{2}} = 4$
  2. Multiply both sides by $\sqrt{2}$ to isolate $|a|$: $|a| = 4\sqrt{2}$

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Tips and Common Mistakes

• Focal Length is Always Positive: Always remember that $p$ represents a distance and must be positive. This is why the absolute value sign is crucial in the perpendicular distance formula and in $\frac{|a|}{\sqrt{2}}$.
• Perpendicular Distance Formula Accuracy: Double-check your substitutions into the perpendicular distance formula. Ensure the line equation is in $Ax+By+C=0$ form.
• Efficiency of Geometric Properties: This problem is a classic example where understanding the geometric properties of a parabola (like the relationship between focus, tangent at vertex, and focal length) is far more efficient than trying to find the general equation of the parabola and then extracting its latus rectum.
• Final Answer Check: The question specifically asks for $|a|$, so ensure your final answer is positive.

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Summary and Key Takeaway

This problem beautifully illustrates how fundamental geometric properties of conic sections can simplify complex-looking problems. The core idea is that the perpendicular distance from the focus to the tangent at the vertex directly gives us the focal length ($p$). Once $p$ is determined, the length of the latus rectum, which is always $4p$, allows us to solve for the unknown parameter. This approach avoids the need to determine the entire equation of the parabola, making the solution quick and elegant.

The final answer is $\boxed{4\sqrt{2}}$.