Problem 137 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

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 (a) \((0,2)\) (b) \((1,0)\) (c) \((0,1)\) (d) \((2,0)\)

Step-by-Step Solution

Answer

The vertex of the parabola is at (1,0).

Step 1: Understanding the Parabola

We are given that the parabola has its focus at the origin, \(0, 0\), and the directrix is the line \ x = 2 \.

Step 2: Determine the Vertex Location

The vertex of the parabola is equidistant from the focus and the directrix. Since the directrix is a vertical line \ x = 2 \, we find the midpoint between the focus \(0, 0\) and the point on the directrix that is horizontally aligned with the focus, which is \(2, 0\). The x-coordinate of this midpoint is \frac{0+2}{2} = 1\. The y-coordinate remains 0, so the vertex is at \(1,0\).

Step 3: Verify by Definition of Parabola

For verification, the point \(1, 0\) should be equidistant from the focus \(0, 0\) and the directrix line \(x = 2\). The distance from \(1, 0\) to the focus is \sqrt{(1-0)^2 + (0-0)^2} = 1\. The distance from \(1, 0\) to the line \(x = 2\) is \|1 - 2| = 1\. Both distances are equal, confirming that \(1, 0\) is indeed the vertex.

Key Concepts

Focus and Directrix

The focus and directrix are fundamental components that define the shape and position of a parabola. In this case, we are given that the parabola has its focus at the origin, which is the point \(0, 0\). This point serves as the focal point where rays of light or paths of particles converge or disperse within the curve of the parabola.
What about the directrix? The directrix is a line that the parabola "leans against," influencing its orientation and opening. Here, the directrix is the vertical line \(x = 2\). This line helps to determine the overall structure and positioning of the parabola, even though it doesn't seem to visibly intersect with the curve itself.

• The focus acts as a central guiding point, pulling or pushing the curve of the parabola toward it.
• The directrix, on the other hand, acts as a balancing force, maintaining the shape of the parabola on the opposite side of the vertex.

Understanding how the focus and directrix interact is key to grasping how a parabola is formed and why it looks the way it does.

Vertex of Parabola

The vertex of a parabola is a crucial point that lies exactly halfway between the focus and the directrix. It is the point where the parabola changes direction, marking its lowest or highest point, depending on its orientation. In our exercise, we determined the vertex to be at the point \(1, 0\).
To find the vertex, we identify the midpoint between the focus \(0, 0\) and a point directly on the directrix line \(x = 2\) that shares the same vertical position as the focus, which is \(2, 0\).

• The x-coordinate of the midpoint is computed as the average between the focus \(x = 0\) and the directrix point \(x = 2\), resulting in \((0+2)/2 = 1\).
• The y-coordinate remains the same at 0, as both the focus and the directrix point share the same \(y\)-value.

This calculation confirms that \(1, 0\) is indeed the vertex of the parabola, exemplifying the balancing act between the focus and the directrix.

Equidistant Property

The equidistant property is a defining attribute of parabolas. This property states that any point on the parabola is equidistant from the focus and the directrix. It is this property that ensures the symmetric nature of a parabola and aids in precisely determining its geometry.
To verify this property in our example, we test the vertex point \(1, 0\). We calculate:

• The distance from the vertex \(1, 0\) to the focus \(0, 0\) is calculated using the distance formula: \(\sqrt{(1-0)^2 + (0-0)^2} = 1\).
• The distance from the vertex \(1, 0\) to the directrix \(x=2\) is simply the absolute horizontal distance: \(|1 - 2| = 1\).

Both these distances equate to 1, proving the vertex \(1, 0\) retains the equidistant property and confirming it is correctly positioned between the focus and directrix. This equidistant balance is why the parabola arches in its distinctive U-shape, maintaining equal spacing from both the focus and the directrix.