1. Understanding the Core Concepts: The Parabola's DNA

To master problems involving parabolas, it's essential to grasp their fundamental properties and the relationships between their key components. This problem hinges on two critical ideas: the parameter 'a' and the length of the latus rectum.

• The Parameter 'a' (Focal Length): This constant, often denoted as 'a', is the defining characteristic of a parabola's "size" or "width." It represents:

  • The perpendicular distance from the vertex ($V$) to the focus ($F$).
  • The perpendicular distance from the vertex ($V$) to the directrix ($L$). This dual definition is incredibly powerful for problem-solving.

• The Latus Rectum: This is a special chord of the parabola that passes through the focus and is perpendicular to the axis of symmetry. Its length is a fixed characteristic for any given parabola, directly related to 'a': $\text{Length of Latus Rectum} = 4a$

• Perpendicular Distance Formula: To calculate the shortest distance, $d$, from a point $(x_1, y_1)$ to a straight line $Ax + By + C = 0$, we use the formula: $d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$

Our Strategy for This Problem: The question provides us with the vertex and the directrix. Our goal is to find the length of the latus rectum. By definition, the parameter 'a' is the perpendicular distance from the vertex to the directrix. Therefore, our plan is:

1. Calculate 'a' using the perpendicular distance formula.
2. Once 'a' is known, calculate the length of the latus rectum using the formula $4a$.

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2. Step-by-Step Solution with Detailed Explanations

Let's break down the problem into manageable steps, explaining the rationale behind each action.

Step 1: Identify and Organize the Given Information

• We are given the coordinates of the vertex of the parabola: $V = (2, -1)$.
• We are given the equation of the directrix of the parabola: $L: 4x - 3y = 21$.

Why this step is crucial: Clearly identifying what's provided is the foundational step in problem-solving. It helps us map the given data to the relevant formulas. In this case, having the vertex and directrix immediately signals that we should use the perpendicular distance formula to find 'a'.

Step 2: Prepare the Directrix Equation and Vertex Coordinates for the Distance Formula

The perpendicular distance formula requires the line equation to be in the standard form $Ax + By + C = 0$. The given directrix equation is $4x - 3y = 21$. To convert it to the standard form, we simply move the constant term to the left side: $4x - 3y - 21 = 0$ From this equation, we can now precisely identify the coefficients:

• $A = 4$
• $B = -3$
• $C = -21$ (Note the negative sign!)

The coordinates of the vertex, which will serve as our point $(x_1, y_1)$ for the distance calculation, are:

• $x_1 = 2$
• $y_1 = -1$

Why this step is crucial: The perpendicular distance formula is sensitive to the signs of the coefficients $A$, $B$, and $C$. By explicitly rewriting the directrix equation in the $Ax + By + C = 0$ form, we prevent common errors, especially with the sign of $C$. Incorrect identification of these values will lead to an incorrect 'a'.

Step 3: Calculate the Parameter 'a' using the Perpendicular Distance Formula

As per the definition, the parameter 'a' is the perpendicular distance from the vertex to the directrix. We will now apply the perpendicular distance formula using the values identified in Step 2.

$a = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$ Substitute $x_1=2$, $y_1=-1$, $A=4$, $B=-3$, and $C=-21$ into the formula: $a = \frac{|4(2) + (-3)(-1) + (-21)|}{\sqrt{4^2 + (-3)^2}}$ Now, let's perform the arithmetic calculations carefully: $a = \frac{|8 + 3 - 21|}{\sqrt{16 + 9}}$ $a = \frac{|11 - 21|}{\sqrt{25}}$ $a = \frac{|-10|}{5}$ Since distance must always be a non-negative value, we take the absolute value of the numerator: $a = \frac{10}{5}$ $a = 2$

Why this step is crucial: Finding the value of 'a' is the central intermediate step. 'a' is the intrinsic parameter that dictates the "shape" of the parabola, and it's directly used to calculate the length of the latus rectum, which is our ultimate goal. Without accurately finding 'a', we cannot proceed to the final answer.

Step 4: Calculate the Length of the Latus Rectum

With the value of $a$ now determined, we can use the standard formula for the length of the latus rectum.

• Length of Latus Rectum $= 4a$
• Substitute the calculated value $a=2$: $\text{Length of Latus Rectum} = 4(2) = 8$

Why this step is crucial: This step directly answers the question posed in the problem. Having successfully calculated 'a', we simply apply the definition of the latus rectum's length to arrive at the final solution.

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3. Important Tips and Common Mistakes to Avoid

• Master the Definition of 'a': Always remember that 'a' is the distance from the vertex to the focus AND the distance from the vertex to the directrix. This understanding is foundational for solving most parabola problems.
• Absolute Value is Crucial: When using the perpendicular distance formula, never forget the absolute value bars in the numerator, i.e., $|Ax_1 + By_1 + C|$. Distance is a scalar quantity and must always be non-negative. Forgetting this can lead to incorrect or even negative values for 'a', which is physically impossible for a distance.
• Standard Form of Line Equation is Key: Always convert the line equation (directrix in this case) to the standard form $Ax + By + C = 0$ before identifying $A$, $B$, and $C$. A very common mistake is to use $C=21$ instead of $C=-21$ if the equation is left as $4x - 3y = 21$. This small sign error will propagate through your calculations.
• Distinguish 'a' from '4a': Be careful not to confuse the parameter 'a' (the focal length) with the length of the latus rectum, which is $4a$. The question explicitly asks for the length of the latus rectum, not just the parameter 'a'.
• Geometric Visualization: While not strictly necessary for this calculation, try to visualize the parabola. The vertex is the turning point, the directrix is a line outside the parabola, and the focus is a point inside. The latus rectum is a segment through the focus, parallel to the directrix. This mental image can help reinforce the concepts.

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4. Summary and Key Takeaway

This problem is a quintessential example of how fundamental definitions and formulas in coordinate geometry are applied to conic sections. The core idea was to leverage the definition of the parabola's parameter 'a' as the perpendicular distance from its vertex to its directrix. By accurately applying the perpendicular distance formula, we successfully determined 'a', and subsequently, the length of the latus rectum ($4a$). This problem reinforces the importance of knowing the geometric properties of parabolas and the precise application of distance formulas.

The final answer is $\boxed{8}$.