Problem 32 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The shortest distance between the parabolas \(y^{2}=4 x\) and \(y^{2}=2 x-6\) is (A) 2 (B) \(\sqrt{5}\) (C) 3 (D) none of these

Step-by-Step Solution

Answer

The shortest distance is 3 units.

Step 1: Identify the Parabolas

The two given parabolas are described by the equations:1. \( y^2 = 4x \)2. \( y^2 = 2x - 6 \). The first parabola is a standard parabola opening to the right, and the second also opens to the right but is shifted horizontally and/or vertically. Both have axes of symmetry parallel to the x-axis.

Step 2: Convert the Second Parabola

The second equation \( y^2 = 2x - 6 \) can be rewritten in a standard form. First, solve for \( x \):\( x = \frac{y^2 + 6}{2} \). This rearranges the equation to show that the parabola is horizontally shifted.

Step 3: Find the Vertices

For the first parabola, \( y^2 = 4x \), the vertex is at the point (0,0). For the second parabola, \( x = \frac{y^2}{2} + 3 \), the vertex is at the point (3,0), indicating that it has moved 3 units to the right compared to the first parabola.

Step 4: Calculate the Distance

Since the vertices are (0,0) and (3,0), the distance between these two points is simply the distance along the x-axis. This distance is calculated as:\( \sqrt{(3 - 0)^2 + (0 - 0)^2} = \sqrt{3^2} = 3 \).

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a right circular cone with a plane. Depending on the angle and position of the intersection, you get different types of curves. These include circles, ellipses, parabolas, and hyperbolas. Each type has its own unique set of properties and equations that define its shape and position in a coordinate plane.
Parabolas are one of the most common conic sections, often found in nature and used in various applications such as satellite dishes and headlights. A key feature of a parabola is its symmetry and the fact that it can be defined by a quadratic equation. Understanding how these different conic sections relate to each other and their properties helps in solving geometric and real-world problems.

• Circles: All points are equidistant from a center point.
• Ellipses: Similar to a stretched circle, with two focal points.
• Parabolas: Defined by a point (focus) and a line (directrix).
• Hyperbolas: Two symmetric open curves, each approaching an asymptote.

Standard Form of Parabola

The standard form of a parabola is fundamental in determining its properties and understanding its orientation and position. A parabola that opens horizontally can be expressed in the form: \[ y^2 = 4px \] Where \( p \) is the distance from the vertex to the focus. If the parabola opens vertically, it is typically given as: \[ x^2 = 4py \] In either case, these equations allow us to identify the vertex and the focus very easily.
A key transformation applied to parabolas is shifting. When a parabola, such as the one described by \( y^2 = 2x - 6 \), is rewritten to show it in standard form, it reveals how it has moved from its original position. The given second parabola rearranges as \( x = \frac{y^2}{2} + 3 \), showing a horizontal shift 3 units to the right. Understanding these transformations is crucial in analyzing and comparing different parabolas.

Axis of Symmetry

The axis of symmetry is an imaginary line that runs through the vertex of the parabola and divides it into two identical mirror-image halves. This line is crucial for identifying points of interest and symmetry-related properties of the parabola.
In standard parabolas like those given in the exercise, the axis of symmetry is parallel to one of the coordinate axes. For parabolas opening horizontally, such as \( y^2 = 4x \) and \( y^2 = 2x - 6 \), the axis of symmetry is parallel to the x-axis.
Knowing the axis of symmetry allows us to quickly determine other properties of the parabola, such as whether two parabolas can be symmetric with respect to each other or exactly overlap. Recognizing the axis of symmetry aids in visualizing geometry problems and understanding the underlying algebraic solutions.
In practical applications, this line helps ensure that structures or paths are balanced and optimal.