Straight LineHardBloom L3
Question
If $L$ is the line whose equation is $ax + by = c$. Let $M$ be the reflection of $L$ through the $y$-axis, and let $N$ be the reflection of $L$ through the $x$-axis. Which of the following must be true about $M$ and $N$ for all choices of $a,b$ and $c$ ?
Options
A.The $x$-intercepts of $M$ and $N$ are equal.
B.The $y$-intercepts of $M$ and $N$ are equal.
C.The slopes of $M$ and $N$ are equal.
D.The slopes of $M$ and $N$ are reciprocal.
Solution
Sol. [Key Concept: Reflecting a line in the axes preserves $|$slope$|$ ⇒ the two images have equal slope.]
Reflecting a graph over the $x$-axis results in the line $M$ whose equation is $ax - by = c$, while a reflection through the $y$-axis results in the line $N$ whose equation is $- ax + by = c$. Both clearly have slope equal to $a/b$ (from, say, the slope-intercept form of the equation.)
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