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Axis of reflection

Mathematics Asked by Ham Tesh on December 25, 2021

Is there any way to get the equation of axis of reflection given two intersecting lines without sketching?

Example question: The image of the line $p:; y-2x=3$ is the line $q:;2y-x=9$. Find the equation of axis of reflection?

3 Answers

I use the given numerical values to demonstrate the logic.

  1. Find X(h, k), the point of intersection of p and q. X = (1, 5).

enter image description here

2.1) Let Q be the point that q cuts the x-axis. $Q = ... = (-9, 0)$.

2.2) Similarly, from p, we have $P = ... (-1.5, 0)$

  1. Find XQ and XP. Note that $dfrac {XQ}{XP} = 2 : 1$ (an accurate approimate).

  2. Let R = (?, 0) be the point that the required line cut the x-axis. By angle bisector theorem, $dfrac {XQ}{XP} = dfrac {QR}{RP} = dfrac {2}{1}$

4.1) Note that QR + RP = QP = 7.5. Then, R = ... = (-4, 0).

  1. Obtain the required by two point form.

Answered by Mick on December 25, 2021

Rewrite your two lines $p,q$ in parametric form with unit speed as

$$p(t) = frac1{sqrt{5}}(1,2)t+(1,5), quad q(t) = frac1{sqrt{5}}(2,1)t+(1,5)$$ for $t in Bbb{R}$ with $S =(1,5)$ being their intersection point.

Now for any $t in Bbb{R}$ the points $p(t)$ and $q(pm t)$ are equally far from $S$ since $$|S-p(t)| = left|frac1{sqrt{5}}(1,2)(t)right| =|t| = left|frac1{sqrt{5}}(2,1)(pm t)right| = |S-q(pm t)|.$$ Therefore if the $r_1,r_2$ are two reflexion lines, we have that $r_1$ and $r_2$ contain the midpoints of the points, say, $p(t)$ and $q(pm t)$ so we can calculate them as $$r_1(t) = frac{p(t)+q(t)}2 = frac3{2sqrt{5}}(1,1)t+(1,5) implies x-y+4=0,$$ $$r_2(t) = frac{p(t)+q(-t)}2 = frac1{sqrt{5}}(-1,1)t+(1,5) implies x-y-6=0.$$

Answered by mechanodroid on December 25, 2021

Say $ell$ is that line (notice that we have actually 2 such lines). Then $ell$ is angle bisector for that twp lines. Now point $T(x,y)$ is on this bisector iff $d(T,p)=d(T,q)$.

So in your case $${|-2x+y-3|over sqrt{5}} = {|-x+2y-9|over sqrt{5}}$$

Now solve this equation and you will get (two) solution.

Answered by Aqua on December 25, 2021

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