Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

Let the mirror image of the point $P(-1,2,-4)$ in the plane $x-y-2z+1=0$ be $Q(x_1,y_1,z_1)$.

The formula for the mirror image of a point $(x_0,y_0,z_0)$ in the plane $ax+by+cz+d=0$ is given by:

$\frac{x_1-x_0}{a}=\frac{y_1-y_0}{b}=\frac{z_1-z_0}{c}=-2\frac{a{x}_0+b{y}_0+c{z}_0+d}{a^2+b^2+c^2}$

Substituting the given values:

$\frac{x_1-(-1)}{1}=\frac{y_1-2}{-1}=\frac{z_1-(-4)}{-2}=-2\left(\frac{1(-1)-1(2)-2(-4)+1}{1^2+(-1{)}^2+(-2{)}^2}\right)$

$\frac{x_1+1}{1}=\frac{y_1-2}{-1}=\frac{z_1+4}{-2}=-2\left(\frac{-1-2+8+1}{1+1+4}\right)$

$\frac{x_1+1}{1}=\frac{y_1-2}{-1}=\frac{z_1+4}{-2}=-2\left(\frac{6}{6}\right)=-2$

Equating each term to $-2$:

$x_1+1=-2\Rightarrow x_1=-3$

$y_1-2=2\Rightarrow y_1=4$

$z_1+4=4\Rightarrow z_1=0$

The mirror image is $(-3,4,0)$.

Answer: $(-3,4,0)$