How do we find the co-ordinates of reflection point. And what's the logic behind?

Question

The point R is the reflection of the point −1, 3 in the line 3y + 2x = 33. Find by calculation the coordinates of R.

Answer 1

Hint: Intersection of two perpendicular lines is the basic logic.

Let the reflected point be $(a,b)$. Then the midpoint $(\frac{a-1}{2},\frac{b+3}{2})$ should lie on the line $2x+3y=33$. This will give you one equation between $a$ and $b$.

Now, for second equation, you should notice that the line joining the original and reflected points will be perpendicular to the given line thus the slope will be $\frac{3}{2}$

Thus your second equation will be $$\frac{3-b}{-1-a}=\frac{3}{2} \tag{2}$$

and first equation $$(a-1) +\frac{3(b+3)}{2}=33 \tag{1}$$

This will give your answer