Question

what is the reflection of point of origin about a line with equation x-2y+2=0

Answer 1

Task. What is the reflection point of origin about a line $a$ with equation $x - 2y + 2 = 0$.

Solution. The normal vector to the line is

$$n = (1, -2).$$

Let $b$ be the line passing through the origin and parallel to $n$. Then $b$ is given by parametric equations:

$$x = t, \qquad y = -2t.$$

The origin corresponds to $t = 0$.

Let us find the intersection point $A$ of $a$ and $b$. For this we substitute $x$ and $y$ into the equation of line $a$:

$$\begin{array}{l} t - 2(-2t) + 2 = 0 \\ t + 4t = -2 \\ 5t = -2 \\ t = -2/5 = -0.4. \end{array}$$

Hence the reflection point of origin about $a$ corresponds to the parameter $2t = 2 \cdot (-0.4) = -0.8$. And so this point is

$$(-0.8, -2 \cdot (-0.8)) = (-0.8, 1.6).$$