Problem 2 - A Complete Resource Book in Mathematics for JEE Main 2017

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

(A) (4,-1) is the correct answer; the ray passes through this point.

Step 1: Identify the Parabola Equation

The given equation of the parabola is \((x - 4)^2 = 4(y + 2)\). This is a standard parabola equation in the form \((x - h)^2 = 4p(y - k)\), where \((h, k)\) is the vertex. The vertex is at \((4, -2)\) and it opens upwards as the squared term \((x - 4)^2\) is on the left-hand side.

Step 2: Determine Focus and Directrix

For the standard form \((x - h)^2 = 4p(y - k)\), the focus is at \((h, k + p)\). Here, \(p = 1\), so the focus is at \((4, -1)\). The directrix is the line \(y = k - p\), thus \(y = -3\).

Step 3: Ray Striking Parallel to y-axis

A ray parallel to the y-axis means it has a direction along the y-axis and it's either moving up or down without changing its x-coordinate. When it strikes the parabola, it will reflect through the focus of the parabola, as this is a property of rays striking a parabolic mirror.

Step 4: Determine the Ray's Path after Reflection

According to the property of parabolic mirrors, any ray parallel to the axis of the parabola (the y-axis here) will reflect through the focus of the parabola. Since the focus is at \((4, -1)\), the ray will pass through this point after reflection.

Step 5: Verify the Correct Answer

After reflection, the ray passes through the focus. Checking against the choices, point \((4, -1)\) matches the point we determined as the intersection of the ray path after reflection. Therefore, the correct answer is option (A) \((4, -1)\).

Key Concepts

Parabola Equation

Understanding the equation of a parabola is crucial when working with reflections on concave mirrors. The equation given in the exercise is \((x - 4)^2 = 4(y + 2)\). This particular form is a "standard form" of a parabola that opens upwards. **Standard Form**: It can be generally described as \((x - h)^2 = 4p(y - k)\), where:

• \((h, k)\) is the vertex of the parabola.
• \(p\) represents the distance from the vertex to the focus of the parabola.

Using this form, we can decipher that the vertex of our given parabola is at \((4, -2)\), and since the term \((x - 4)^2\) is on the left, the parabola opens upwards. Changing the values of these constants affects the orientation and position of the parabola, which is essential when analyzing the trajectory of rays reflecting off a concave mirror.

Focus of a Parabola

The focus of a parabola is a key point that influences how light rays reflect off the shape. For the parabola \((x - 4)^2 = 4(y + 2)\), once in the standard form \((x - h)^2 = 4p(y - k)\), it's evident that:

• The vertex \((h, k)\) is at \((4, -2)\).
• The parameter \(p = 1\) since \(4p = 4\).
• The focus is positioned at \((h, k + p)\), which calculates to \((4, -1)\).

The focus is the point through which all parallel incoming rays will pass after reflecting off the surface of a parabolic mirror. With the focus located at \((4, -1)\), any ray coming parallel to the y-axis will converge through this point. This predictable behavior is what makes parabolic mirrors useful in various optical applications, including telescopes and satellite dishes.

Ray Reflection

When rays of light strike a concave mirror shaped like a parabola, such as the one in the exercise, they behave in an orderly manner due to the properties of the parabola. If a ray comes parallel to the y-axis, it will reflect through the focus of the parabola:

• The ray strikes the surface, maintaining the x-coordinate, as it is parallel to the y-axis.
• Upon reflection, it passes through the focus, acting according to the laws of optics.

This behavior occurs because each ray is essentially "guided" by the shape of the parabola to move toward the shared point known as the focus. Hence, when handling reflections in parabolic mirrors, you can anticipate that any parallel incoming light will reflect ideally, meeting at the focus, in this case at the point \((4, -1)\). This important reflection property not only aids in exercises but is practical in real-world applications to focus light or sound effectively.