A parabola has the origin as its focus and the line x = 2 as the directrix. Then the vertex of the p

Q: Wait, why is the axis of symmetry $y=0$?

Great question! The directrix is the vertical line $x=2$. The axis of symmetry is always perpendicular to the directrix and must pass through the focus $(0,0)$. A line perpendicular to a vertical line is horizontal. Since it passes through $(0,0)$, its equation is $y=0$.

Q: Why is the vertex always the midpoint of the focus and the intersection point?

By definition, the vertex is the point on the parabola closest to the directrix. Since the distance from the focus to the directrix is $d$, the vertex must be at a distance $d/2$ from both. Thus, it is exactly the midpoint of the segment connecting the focus and the intersection point $Z$.

Q: How do I know which way the parabola opens?

Look at the relative positions. The focus is at $(0,0)$ and the directrix is at $x=2$. The parabola must 'wrap around' the focus and open away from the directrix. Since the directrix is to the right of the focus, the parabola opens to the left.

Question

A parabola has the origin as its focus and the line $x = 2$ as the directrix. Then the vertex of the parabola is at

Options:

(A) $(0, 2)$

(B) $(1, 0)$

(C) $(0, 1)$

(D) $(2, 0)$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future engineer! Today, we are exploring the elegant architecture of a parabola. Imagine you are standing on a coordinate plane with a focus at the origin, $F(0,0)$.

To your right, standing like a wall, is the vertical line $x=2$, which serves as the directrix. A parabola is defined as the set of all points that are equidistant from the focus and the directrix.

### The Axis of Symmetry: Finding Our Bearing

Before we find the vertex, we must identify the axis of symmetry. Think of this as the spine of the parabola.

By definition, the axis of symmetry must pass through the focus and be perfectly perpendicular to the directrix. Since our directrix is the vertical line $x=2$, our axis of symmetry must be a horizontal line.

Because it must pass through the focus at $(0,0)$, the equation of this axis is simply $y=0$, which is the x-axis itself. This serves as our baseline.

### The Intersection Point $Z$

Now, let us find the point where this axis of symmetry meets the directrix. Let us call this point $Z$.

Since $Z$ lies on the vertical line $x=2$ and the horizontal line $y=0$, its coordinates are clearly $Z(2,0)$. This point $Z$ is crucial because it helps us define the "width" of our parabola's opening.

### The Midpoint Property: The Heart of the Parabola

The vertex $V$ of a parabola is the point where the curve is closest to the directrix. Because the parabola is perfectly symmetric, the vertex must lie exactly halfway between the focus $F$ and the intersection point $Z$.

To find the coordinates of the vertex, we use the midpoint formula:

$$V = \left( \frac{x_F + x_Z}{2}, \frac{y_F + y_Z}{2} \right)$$

### Final Calculation

Let us plug in our values. We have the focus $F(0,0)$ and the point $Z(2,0)$.

The x-coordinate of the vertex is:

$$\frac{0 + 2}{2} = 1$$

The y-coordinate is:

$$\frac{0 + 0}{2} = 0$$

Therefore, the vertex of our parabola is at **$V(1,0)$**.

You have successfully navigated the geometry of the parabola. Remember, in JEE, visualization is your greatest asset; when you see a focus and a directrix, visualize the symmetry, find the axis, and let the midpoint property guide you home.

Animated Solution for Mathematics - Conic Sections: A parabola has the origin as its focus and the line $x = 2$ as the directrix. Then the vertex of the parabola is at

Select Answer:

Visualized Solution

Identify Focus and Directrix

Determine the Axis of Symmetry

Equation of the Axis

Find Intersection Point $Z$

The Vertex Property

Midpoint Formula Setup

Substitute Coordinates

Calculate X-coordinate

Calculate Y-coordinate

Final Vertex and Parabola

The Sigma Insight: Standard Equations of Parabola, Ellipse, and Hyperbola

Analyzing the Setup

The Axis of Symmetry

The Intersection Point $Z$

The Midpoint Property

Final Calculation

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Animated Solution for Mathematics - Conic Sections: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A parabola has the origin as its focus and the line x=2 as the directrix. Then the vertex of the parabola is at