Question

Equation of the image of the pair of rays $y = |x|$ by the line $x = 1$ is

A) $|y| = x + 2$

B) $|y| + 2 = x$

C) $y = |x - 2|$

D) none of these

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Answer

To solve this problem, we start by considering the original pair of rays represented by the equation $$y = |x|.$$ This equation can be decomposed into two linear parts:

  1. $$y = x \quad \text{(when } x \geq 0\text{)}$$
  2. $$y = -x \quad \text{(when } x < 0\text{)}$$

Next, we find the image of these rays reflected across the line $$x = 1.$$

For ray ( y = x ):

  • Consider a point $(a, a)$ on this ray. The mirror image across $x = 1$ will be located at $(2 - a, a)$.

For ray ( y = -x ):

  • Consider a point $(a, -a)$ on this ray. The mirror image will be at $(2 - a, -a)$.

Observing the coordinates of the images, we see that:

  • When $a > 1$, $(2 - a, a)$ becomes $(b, c)$ where $c = b - 2$.
  • When $a < 1$, $(2 - a, -a)$ transforms to $(b, c)$ where $c = 2 - b$.

We combine these transformations into a modulus function that dictates the reflection behavior for all values of $x$. This leads us to: $$ y = |x - 2| $$ which captures both transformations such that when $x < 2$, $y$ takes on positive slopes that descend, and when $x > 2$, $y$ takes on negative slopes.

Conclusion: The correct equation that represents the image of the pair of rays $y = |x|$ by the line $x = 1$ is $$ \boldsymbol{y = |x - 2|} $$

Thus, the correct option is C) $y = |x - 2|$.


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