Fundamental Concepts of a Parabola and its Vertex

A parabola is a fundamental conic section defined as the locus of a point that moves in a plane such that its distance from a fixed point (called the focus) is always equal to its distance from a fixed line (called the directrix).

To understand a parabola and its properties, we identify key components:

• Focus (S): The fixed point that defines the parabola.
• Directrix: The fixed line that defines the parabola.
• Axis of the Parabola: This is the line that passes through the focus and is perpendicular to the directrix. It serves as the axis of symmetry for the parabola.
• Vertex (V): The point on the parabola that lies on its axis. It is the point on the parabola closest to both the focus and the directrix.

Crucial Property for the Vertex: One of the most efficient ways to locate the vertex of a parabola, given its focus and directrix, is by using the following property: The vertex (V) of a parabola is the midpoint of the focus $S$ and the point where the axis of the parabola intersects the directrix. Let's denote this intersection point as $M$. Therefore, $V$ is the midpoint of the segment $SM$. This property is the cornerstone for solving this problem.

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Problem Statement We are given a parabola with the following characteristics:

• Its focus is at the origin, $S = (0,0)$.
• Its directrix is the line with the equation $x=2$.

Our objective is to find the coordinates of the vertex of this parabola.

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Step-by-Step Solution

Step 1: Identify the Given Focus and Directrix

• Why this step is taken: Clearly stating the given information is the first and most crucial step in any problem-solving process. It sets the foundation for subsequent calculations.
• From the problem statement:
  • Focus (S): The origin, $S = (0,0)$.
  • Directrix: The line $x=2$.

Step 2: Determine the Axis of the Parabola

• Why this step is taken: To apply the midpoint property for the vertex, we need two points: the focus $S$ and the point $M$ where the axis intersects the directrix. To find $M$, we first need the equation of the axis of the parabola.
• Recall the definition: The axis of the parabola passes through the focus and is perpendicular to the directrix.
• Our directrix is $x=2$. This is a vertical line (parallel to the y-axis).
• Since the axis must be perpendicular to a vertical line, the axis must be a horizontal line (parallel to the x-axis).
• Furthermore, this horizontal axis must pass through the focus $S(0,0)$.
• A horizontal line passing through $(0,0)$ has the equation $y=0$. This is the x-axis.
• Therefore, the axis of the parabola is the x-axis, with equation $y=0$.

Step 3: Find the Point of Intersection (M) of the Axis and the Directrix

• Why this step is taken: We now have the focus $S$ and need the point $M$ to use the midpoint formula. Point $M$ is defined as the intersection of the axis and the directrix.
• We have:
  • Equation of the axis: $y=0$
  • Equation of the directrix: $x=2$
• To find their intersection point $M$, we simply use these two equations. The x-coordinate of $M$ is $2$ (from the directrix equation), and the y-coordinate of $M$ is $0$ (from the axis equation).
• Thus, the point of intersection is $M = (2,0)$.

Step 4: Calculate the Coordinates of the Vertex (V)

• Why this step is taken: We have successfully identified both the focus $S$ and the point $M$ (where the axis meets the directrix). We can now directly apply the fundamental property that the vertex $V$ is the midpoint of the segment $SM$.
• We have:
  • Focus $S = (x_1, y_1) = (0,0)$
  • Point $M = (x_2, y_2) = (2,0)$
• Using the midpoint formula, which states that for two points $(x_1, y_1)$ and $(x_2, y_2)$, their midpoint is $\left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right)$: $V = \left( \frac{0+2}{2}, \frac{0+0}{2} \right)$ $V = \left( \frac{2}{2}, \frac{0}{2} \right)$ $V = (1,0)$
• Therefore, the vertex of the parabola is at $(1,0)$.

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Visualisation and Understanding the Parabola's Orientation

It's always beneficial to visualize the parabola to confirm our result and understand its orientation:

• The focus $S$ is at $(0,0)$.
• The directrix is the vertical line $x=2$.
• The axis of the parabola is the x-axis ($y=0$).
• Our calculated vertex $V$ is at $(1,0)$.
• Observe that the focus $(0,0)$ is to the left of the directrix $x=2$. This implies that the parabola must open towards the focus, i.e., to the left (in the negative x-direction).
• The vertex $(1,0)$ lies exactly in the middle of the focus and the directrix, which is consistent. The distance from the vertex to the focus is $a = |1-0|=1$, and the distance from the vertex to the directrix is $|1-2|=1$. This confirms the vertex's position and the value of $a=1$.
• The standard equation for a parabola opening to the left with vertex $(h,k)$ is $(y-k)^2 = -4a(x-h)$. Substituting $V=(1,0)$ and $a=1$, we get $(y-0)^2 = -4(1)(x-1)$, or $y^2 = -4(x-1)$.

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Common Pitfalls and Expert Tips

1. Incorrect Axis Identification: Always ensure the axis is perpendicular to the directrix AND passes through the focus. If the directrix is $x=k$ (vertical), the axis is $y=y_{focus}$ (horizontal). If the directrix is $y=k$ (horizontal), the axis is $x=x_{focus}$ (vertical).
2. Misidentifying the "Other Point" (M): The vertex is the midpoint of the focus and the intersection of the axis with the directrix, not just any arbitrary point on the directrix. This distinction is crucial.
3. Midpoint Formula Errors: Double-check your arithmetic when applying the midpoint formula, especially if negative coordinates are involved.
4. Orientation Confusion: A quick sketch of the focus and directrix helps visualize the parabola's opening direction. The parabola always "wraps around" its focus and opens away from its directrix. This can help you verify if your calculated vertex makes sense.
5. Distance 'a': The distance from the vertex to the focus is equal to the distance from the vertex to the directrix. This distance is typically denoted by 'a'. For our problem, $V=(1,0)$, $S=(0,0)$, Directrix $x=2$. Distance $VS = \sqrt{(1-0)^2+(0-0)^2} = 1$. Distance from $V$ to $x=2$ is $|1-2|=1$. This consistency confirms the vertex's location.

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Summary and Key Takeaway

To efficiently find the vertex of a parabola given its focus and directrix, the most direct and reliable method is to utilize the property that the vertex is the midpoint of the focus and the point where the axis of the parabola intersects the directrix. By systematically identifying the focus, determining the axis, finding the intersection point on the directrix, and applying the midpoint formula, we successfully located the vertex at $(1,0)$. This corresponds to option (B). This method is robust and avoids the more complex algebraic derivation of the parabola's full equation.