A plane mirror along the x axis and one end of the mirror is at the origin find the equation of image of an object which is in the form of sine curve given by Y equal a sin kx

To find the equation of the image of an object represented by the sine curve \( y = a \sin(kx) \) when reflected in a plane mirror along the x-axis, we need to understand how reflection works in this context. The plane mirror will invert the vertical position of any point on the sine curve while keeping the horizontal position unchanged.

Understanding Reflection in a Plane Mirror

When an object is reflected in a plane mirror, the coordinates of the points on the object change according to the following rules:

• The x-coordinate remains the same.
• The y-coordinate changes sign.

Applying Reflection to the Sine Curve

Given the sine curve \( y = a \sin(kx) \), we can denote any point on this curve as \( (x, a \sin(kx)) \). When this point is reflected across the x-axis, the new coordinates of the image point will be \( (x, -a \sin(kx)) \).

Equation of the Image Curve

Thus, the equation of the image of the sine curve after reflection in the mirror along the x-axis can be expressed as:

Image Equation: \( y = -a \sin(kx) \)

Visualizing the Transformation

To visualize this transformation, consider the following:

• The original sine curve oscillates above and below the x-axis, reaching a maximum of \( a \) and a minimum of \( -a \).
• Upon reflection, the peaks of the original sine curve (where \( y = a \)) become troughs (where \( y = -a \)), and the troughs (where \( y = -a \)) become peaks (where \( y = a \)).

Example for Clarity

Let’s take a specific example where \( a = 2 \) and \( k = 1 \). The original sine curve is:

Original Curve: \( y = 2 \sin(x) \)

After reflection, the image curve will be:

Image Curve: \( y = -2 \sin(x) \)

This means that every point on the original curve is mirrored across the x-axis, effectively flipping it upside down.

Conclusion

In summary, reflecting the sine curve \( y = a \sin(kx) \) in a plane mirror along the x-axis results in the image curve \( y = -a \sin(kx) \). This transformation is a straightforward application of the properties of reflection, where the vertical coordinates are inverted while the horizontal coordinates remain unchanged.