locus of points such 3 points are collinear, in space (Chile 2016 L2 P6)

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locus of points such 3 points are collinear, in space (Chile 2016 L2 P6)

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#1 h May 31, 2019, 10:03 PM • 2 Y

Y by Adventure10, Mango247

Let $P_1$ and $P_2$ be two non-parallel planes in space, and $A$ a point that does not It is in none of them. For each point $X$ , let $X_1$ denote its reflection with respect to $P_1$ , and $X_2$ its reflection with respect to $P_2$ . Determine the locus of points $X$ for the which $X_1, X_2$ and $A$ are collinear.

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#2 h Oct 13, 2020, 7:19 PM

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Plane $P_{1}\ :\ z=0$ , plane $P_{2}\ :\ y=0$ .
Point $A(a,b,c)$ .

Point $X(k,l,m)$ , then $X_{1}(k,l,-m)$ and $X_{1}(k,-l,m)$ .

Line $X_{1}X_{2}\ :\ x=k$ and $\frac{y-l}{2l}=\frac{z+m}{-2m}$ .
Point $A \in X_{1}X_{2}\ :\ a=k$ and $\frac{b-l}{l}=\frac{c+m}{-m}$ .

Conclusion: point $X(a,l,-\frac{c \cdot l}{b})$ .
Point $X$ lies on a line $\left\{\begin{array}{ll} x=a \\ cy+bz = 0 \end{array}\right.$ .

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