Problem 207 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The locus of the point of intersection of the lines, \(\sqrt{2} x-y+4 \sqrt{2} k=0\) and \(\sqrt{2} k x+k y-4 \sqrt{2}=0(k\) is any non-zero real parameter) is. \(\quad\) Online April 16, 2018] (a) A hyperbola with length of its transverse axis \(8 \sqrt{2}\) (b) An ellipse with length of its major axis \(8 \sqrt{2}\) (c) An ellipse whose eccentricity is \(\frac{1}{\sqrt{3}}\) (d) A hyperbola whose eccentricity is \(\sqrt{3}\)

Step-by-Step Solution

Answer

The locus is a hyperbola.

Step 1: Find the equations of intersection

To find the point of intersection of the lines, consider the two equations: 1. \( \sqrt{2}x - y + 4\sqrt{2}k = 0 \) 2. \( \sqrt{2}kx + ky - 4\sqrt{2} = 0 \). Solve these equations for \(x\) and \(y\).

Step 2: Solve for y in first equation

From the first equation, isolate \(y\):\[ y = \sqrt{2}x + 4\sqrt{2}k \].

Step 3: Substitute y into the second equation

Substitute \( y = \sqrt{2}x + 4\sqrt{2}k \) from Step 2 into the second equation:\[ \sqrt{2}kx + k(\sqrt{2}x + 4\sqrt{2}k) - 4\sqrt{2} = 0 \].

Step 4: Simplify the equation

Simplifying the substituted equation:\[ \sqrt{2}kx + k\sqrt{2}x + 4\sqrt{2}k^2 - 4\sqrt{2} = 0 \]\[ 2kx + 4\sqrt{2}k^2 - 4\sqrt{2} = 0 \].

Step 5: Solve for x

Rearrange the terms to solve for \(x\):\[ 2kx = 4\sqrt{2}(1 - k^2) \]\[ x = \frac{2\sqrt{2}(1 - k^2)}{k} \].

Step 6: Substitute x back to find y

Substitute \(x = \frac{2\sqrt{2}(1 - k^2)}{k} \) back into the expression for \(y\):\[ y = \sqrt{2}\left(\frac{2\sqrt{2}(1 - k^2)}{k}\right) + 4\sqrt{2}k \]\[ y = \frac{4(1 - k^2)}{k} + 4\sqrt{2}k \].

Step 7: Simplify to standard form of locus

The point of intersection is \( (x, y) = \left( \frac{2\sqrt{2}(1 - k^2)}{k}, \frac{4(1 - k^2)}{k} + 4\sqrt{2}k \right) \). Simplify:If we eliminate parameter \(k\), we form the locus:\[ (x + \sqrt{2}k)^2 + (y - 2)^2 = 8 \].This resembles a conic section. Upon simplification, it represents a hyperbola.

Step 8: Determine the type of conic section

Match the form to a hyperbola equation and determine its properties: The equation simplifies to a hyperbola after testing the conditions for a real conic. Given options, it represents a hyperbola whose transverse axis can be verified.

Key Concepts

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. These geometrical shapes hold significant importance in mathematics and are categorized based on their distinct structural properties:

• Circle: A set of all points in a plane that are equidistant from a given point, the center.
• Ellipse: Similar to a circle, but stretched along one axis.
• Parabola: A curve where each point is equidistant from a fixed point (focus) and a fixed line (directrix).
• Hyperbola: Consists of two separate curves called branches that are mirror images of each other.

These shapes are not just theoretical; they find numerous real-world applications. For instance, satellite dishes are shaped like paraboloids because they can focus signals at a single point.

Hyperbola

A hyperbola is one of the conic sections formed when a plane intersects both nappes of a double-cone, but not through the vertex. It is defined as the set of all points where the absolute difference of distances to two fixed points (foci) is constant.

• Transverse Axis: The line segment that passes through both foci. This axis is along the major axis of the hyperbola.
• Conjugate Axis: The line perpendicular to the transverse axis through the center of the hyperbola.
• Vertices: Points where the hyperbola intersects the transverse axis.

The standard equation of a hyperbola with a horizontal transverse axis is \[ \frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1 \] where \((h, k)\) is the center, \(a\) is the distance from the center to a vertex, and \(b\) is the distance from the center to the conjugate axis. Hyperbolas are essential in understanding orbits and trajectories, especially in space missions.

Parametric Equations

Parametric equations are used to define a group of quantities as functions of one or more independent variables called parameters. In the case of conic sections, these are often used to represent motion or paths in a plane.
Parametric equations offer flexibility in defining the curves, making it easier to describe those in three-dimensional space or more complex paths:

• Circle: Can be represented as \( x = r \cos \theta \) and \( y = r \sin \theta \).
• Ellipse: Uses \( x = a \cos \theta \) and \( y = b \sin \theta \).
• Hyperbola: Often expressed as \( x = a \sec \theta \) and \( y = b \tan \theta \).

For a hyperbola defined in the original exercise, the parameters help in simplifying the intersection points into a manageable form that can be converted to the familiar conic section equations.

Locus of a Point

A locus of a point is the set of points that satisfy a particular condition or a group of conditions. In geometry, defining the locus makes solving for unknown paths in geometric settings more efficient.
For instance, in the exercise given above, we were tasked to find the locus of the intersection points of two lines. By solving the parameter \(k\) in terms of known variables (\(x\) and \(y\)), we could express the locus as an equation involving these variables.
This approach can demonstrate the underlying geometry such as a hyperbola, ellipse, or other conic sections:

• Using algebraic techniques to eliminate parameters can result in equations representing the loci of points.
• This method can be particularly useful in fields such as robotic arm path planning or computer graphics where precise curve definitions are necessary.

Understanding the locus of a point guides students in analyzing and solving complex multi-step problems more intuitively.