Problem 94 - A Complete Resource Book in Mathematics for JEE Main

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

(B) \((1, 0)\)

Step 1: Identify the given elements

The problem states that the focus of the parabola is at the origin, which is the point \((0, 0)\). The directrix is the line \(x = 2\).

Step 2: Determine the vertex location

The vertex of a parabola is located midway between the focus and the directrix. Since the focus is at \((0, 0)\) and the directrix is the line \(x = 2\), the vertex lies on the horizontal line that is halfway between \(x = 0\) and \(x = 2\).

Step 3: Calculating the midpoint

To find the midpoint in terms of \(x\)-coordinates for the vertex, calculate the average of \(0\) and \(2\): \(x_{mid} = \frac{0+2}{2} = 1\). Thus, the vertex is \((1, 0)\) with an \(x\)-coordinate of \(1\) and a \(y\)-coordinate of \(0\).

Step 4: Verify and Conclude

The calculated vertex \((1, 0)\) is consistent with the geometry of a parabola with a vertical directrix. Therefore, it matches with option (B).

Key Concepts

Focus of a Parabola

The focus of a parabola is a critical point that helps define its unique shape. In our example, the focus is located at the origin, which is the point \((0, 0)\). The focus is always located inside the parabola. This means that it sits on the side of the parabolic curve that is away from the directrix.
The proximity and orientation of the focus to the parabola influence how the curve is shaped and directed.

• A key property is that each point on the curve is equidistant from the focus and the directrix.
• Here, the focus being at the origin suggests that the parabola will open based on its relationship to the directrix.

Understanding the focus is crucial as it plays a fundamental role in the equations of parabolas and in the calculations of the other elements such as the vertex.

Directrix of a Parabola

The directrix is a straight line that works in conjunction with the focus to define a parabola. In this problem, the directrix is the vertical line represented as \(x = 2\).
A vital characteristic of the directrix is that each point on the parabola is equally distant from the directrix and the focus. This balance of distances is what shapes the parabolic curve.

• The directrix is always located on the opposite side of the vertex relative to the focus.
• It acts as a boundary that the parabola does not cross. Instead, it mirrors the focus on the opposite side of the curve.

This unique tandem between the focus and the directrix is what helps to accurately position the vertex, which is the midpoint between these two elements.

Vertex of a Parabola

The vertex is the turning point of the parabola, often thought of as the "tip" or "peak." In our example, we determined that the vertex is located at \((1, 0)\). The vertex lies exactly halfway between the focus and the directrix.

• The calculation in our example involved finding the midpoint between the focus at \((0, 0)\) and the vertical line where the directrix \(x = 2\) is located.
• By averaging the x-values of the focus and the directrix, we determine that the x-coordinate of the vertex is \(1\), giving us the point \((1, 0)\).

Effectively, the vertex not only marks the midpoint geometrically but also acts as a reference for the symmetry of the parabola across its vertical axis. It acts as a key point from where the parabola can be graphically sketched and analyzed.