Question

A thin particle moves from \((0,1)\) and gets reflected upon hitting the x-axis at \((√3,0)\). Then the slope of the reflected line is ?

• A. \(\dfrac{1}{√3}\)
• B. \(\dfrac{-1}{√3}\)
• C. \({√3}\)
• D. \(-{√3}\)

Hint:

\(m=\dfrac{y_2-y_1}{x_2-x_1} \)  use this formula

Answer 1

The Correct Option is B

Given that :

Since the particle gets reflected upon hitting the x-axis, the angle of incidence is equal to the angle of reflection. 

(In other words, the angle the particle makes with the x-axis before hitting the x-axis is the same as the angle it makes with the x-axis after reflection.)

The initial position of the particle is \((0, 1)\) and the point where it gets reflected is \((√3, 0)\). The slope of the line connecting these two points is given by:

\(m=\dfrac{0-1}{√3-0}  =\dfrac{-1}{√3}\)

Hence Slope of the line is \(\dfrac{-1}{√3}\) ..(Ans.)

Answer 2

Given:
Initial Position: The particle starts at (0, 1).
Reflection Point: Upon hitting the x-axis, the particle reflects at \((\sqrt{3}, 0)\).

Calculate Slope:
The slope m between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\(m = \frac{y_2 - y_1}{x_2 - x_1}\)
Substitute the coordinates:
\(m = \frac{0 - 1}{\sqrt{3} - 0}\)

Perform Calculation:
Numerically compute the slope: \(m = \frac{-1}{\sqrt{3}}\)

Interpret the Result:
The negative sign indicates that the line slopes downward from left to right.
\(\frac{1}{\sqrt{3}}\) simplifies to \(\frac{\sqrt{3}}{3}\).
The slope of the line connecting the initial position (0, 1) to the reflection point \((\sqrt{3}, 0)\) is \(\frac{-1}{\sqrt{3}}\).

So, the correct option is (B): \(\frac{-1}{\sqrt{3}}\)