Problem 2 - A Complete Resource Book in Mathematics for JEE Main

Question

A ray of light is coming along the line which is parallel to \(y\)-axis and strikes a concave mirror whose intersection with the \(x y\)-plane is a parabola \((x-4)^{2}=4(y+2)\). After reflection, the ray must pass through the point (A) \((4,-1)\) (B) \((0,1)\) (C) \((-4,1)\) (D) none of these

Step-by-Step Solution

Answer

The ray must pass through point (4, -1), corresponding to option (A).

Step 1: Understand the Parabola Equation

The given equation of the parabola is \((x-4)^2 = 4(y+2)\). This is a parabola that opens upwards with its vertex at \((4, -2)\). The focus of the parabola can be calculated using the equation of a parabola \((x-h)^2 = 4p(y-k)\). Here, \(h = 4\), \(k = -2\), and \(4p = 4\), leading to \(p = 1\). Hence, the focus is at \((4, -1)\).

Step 2: Determine the Path of Incoming Ray

The ray of light is said to be coming parallel to the \(y\)-axis. This implies that the line representing the ray is vertical (such as \(x = k\) where \(k\) is constant) and it's moving towards the parabola.

Step 3: Reflection Property of Parabolas

Parabolas have the property that any incoming ray parallel to the axis of symmetry of the parabola reflects through its focus. Thus, a ray parallel to the \(y\)-axis striking this parabola will reflect through the parabola's focus point.

Step 4: Choose the Correct Option

From the properties of the parabola, since the ray after striking the parabola must pass through the focus of the parabola, the correct choice for the reflected ray passing through is point \((4, -1)\), corresponding to option (A).

Key Concepts

Reflection Property of a Parabola

A parabola exhibits a fascinating geometric attribute known as the reflection property. This property states that rays of light initially traveling parallel to the parabola's axis of symmetry will reflect towards its focus.
This occurs because of the unique shape of a parabola, where all points are equidistant from a single point (the focus) and a straight line (the directrix).
In practical terms, when a light ray strikes a parabolic reflector such as the one defined by \[(x-4)^2 = 4(y+2)\], the trajectories of these rays are redirected, converging at the parabola's focus.
This principle is critical in various applications like satellite dishes and headlight reflectors, as it ensures that signals or light are concentrated precisely.

• Light reflects toward the focus.
• Axis-parallel rays converge precisely at one point.
• This principle underlies the working of various optical devices.

By harnessing the reflection property, engineers and designers can manipulate paths of light and signals with high precision.

Focus of Parabola

The focus is a pivotal part of a parabola's geometric structure, serving as one of the defining components which the shape revolves around. For the parabola described by the equation \[(x-4)^2 = 4(y+2)\], the focus is determined through the parameter \(p\) in the standard equation form \[(x-h)^2 = 4p(y-k)\].
Here, \(h = 4\) and \(k = -2\), placing the vertex at \((4, -2)\). Since \(4p = 4\), we find that \(p = 1\). Thus, the focus is located at \((4, -1)\).
In essence:

• The focus serves as the convergence point for incoming parallel rays due to the reflection property.
• It is used in designing parabolic reflectors in antennas and headlights for optimum focus.

Understanding where the focus lies helps in predicting how a parabola will direct rays or signals.

Axis of Symmetry

The axis of symmetry is an essential characteristic of a parabola, dividing it perfectly into two symmetrical halves. For the given parabola \[(x-4)^2 = 4(y+2)\], this line is vertical as the equation is in the form \[(x-h)^2 = 4p(y-k)\].
The axis of symmetry here is the line \(x = 4\), matching the \(x\)-coordinate of the vertex. This line plays a pivotal role because it is along this line that the reflection property operates.

• The parabola is symmetrical around this axis.
• Incident rays parallel to this axis are reflected towards the focus.
• Lighting up the symmetry can be key in applications involving light paths and parabolic reflectors.

The concept of an axis of symmetry is crucial for understanding how the geometric properties of a parabola come together in optical and engineering applications.