Problem 114 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Axis of a parabola lies along \(x\)-axis. If its vertex and focus are at distance 2 and 4 respectively from the origin, on the positive \(x\)-axis then which of the following points does not lie on it? (a) \((5,2 \sqrt{6})\) (b) \((8,6)\) (c) \((6,4 \sqrt{2})\) (d) \((4,-4)\)

Step-by-Step Solution

Answer

Point \((8, 6)\) does not lie on the parabola.

Step 1: Identify Vertex and Focus

The exercise states that the vertex and focus of the parabola lie on the positive x-axis. The vertex is 2 units away from the origin, and the focus is 4 units away. Therefore, the vertex is at \((2, 0)\) and the focus is at \((4, 0)\).

Step 2: Calculate Parabola Equation

For a parabola aligned with the x-axis having vertex \((h, k)\) and focus \((h+p, k)\), the equation is \((y-k)^2 = 4p(x-h)\). Here, \(p = 2\) because the focus is 2 units to the right of the vertex. Substituting \(h = 2\) and \(k = 0\), we get \(y^2 = 8(x-2)\).

Step 3: Substitute Points into the Equation

To find out which point does not lie on the parabola, substitute each point into the equation \(y^2 = 8(x-2)\):- For \((5, 2\sqrt{6})\), calculate: \((2\sqrt{6})^2 = 24\), and \(8(5-2) = 24\). This point lies on the parabola.- For \((8, 6)\), calculate: \(6^2 = 36\), and \(8(8-2) = 48\). This point does not lie on the parabola.- For \((6, 4\sqrt{2})\), calculate: \((4\sqrt{2})^2 = 32\), and \(8(6-2) = 32\). This point lies on the parabola.- For \((4, -4)\), calculate: \((-4)^2 = 16\), and \(8(4-2) = 16\). This point lies on the parabola.

Key Concepts

Vertex

The vertex of a parabola is a significant point that determines its orientation and position on the coordinate plane. In the exercise, the vertex is mentioned as being 2 units away from the origin on the positive x-axis, which places it at the coordinate \( (2, 0) \). This means that the parabola opens horizontally rather than vertically.

Understanding the vertex is crucial for graphing and analyzing parabolas because:

• It serves as the midpoint of the parabola and provides symmetry.
• The vertex helps in determining the direction in which the parabola opens based on the axis alignment (x-axis or y-axis).

The role of the vertex depends on whether the parabola opens along the x-axis or y-axis. In this case, since the exercise specifies that the parabola lies along the x-axis, the vertex coordinates \( (h, k) \) contribute to writing its standard equation. Hence, comprehending the vertex position helps predict the curvature and nature of the parabola.

Focus

The focus of a parabola is another vital component that, along with the vertex, helps determine its orientation and shape. For this problem, the focus is identified at \( (4, 0) \), which is 4 units away from the origin along the x-axis. The distance between the vertex \( (2, 0) \) and the focus \( (4, 0) \) is known as the focal length \( p \), and in this situation, it is 2 units.

The concept of the focus is important because:

• It defines the precise curve of the parabola; every point on the parabola is equidistant from the directrix and the focus.
• The focal distance \( p \) helps determine how "wide" or "narrow" the parabola will be.

When the focus and vertex are understood in the context of the parabola's axis, it becomes easier to visualize and formulate the correct parabola equation. The given exercise shows that understanding the focus's role is crucial in confirming which points lie on the parabola.

Parabola Equation

A parabola's equation represents the specific relationship between its points in a coordinate system, and for this exercise, it lies along the x-axis, resulting in a horizontal orientation.

For a horizontally oriented parabola with a vertex \( (h, k) \) and a focus \( (h+p, k) \), the standard equation is \( (y-k)^2 = 4p(x-h) \). Here, the focal length \( p = 2 \), which translates the problem into deriving its equation: \( y^2 = 8(x-2) \).

Key points about the parabola equation include:

• The parabolic equation transforms with the inclusion of the vertex \( (h, k) \), dictating shifts and opening direction.
• Understanding the standard format of the parabola equation can simplify the process of checking whether specific points are on the parabola by substituting their coordinates into the equation.

Hence, the equation \( y^2 = 8(x-2) \) demonstrates the set of points \( (x, y) \) that comply with the parabola's definition as per distances to the focus and directrix. By substituting various points from the options given in the exercise, one can determine the validities of whether they lie on the parabola or not.