Reflection maps each point to its mirror image across a given line or point. The standard reflection formulas cover the most common JEE scenarios.

Across coordinate axes: x-axis maps (a,b) to (a,-b), y-axis maps (a,b) to (-a,b). Through the origin: (a,b) maps to (-a,-b).

Across the lines y=x and y=-x: (a,b) maps to (b,a) and (-b,-a) respectively. These are equivalent to swapping coordinates (with sign changes for y=-x).

Across vertical/horizontal lines: x=c maps (a,b) to (2c-a,b). y=c maps (a,b) to (a,2c-b). The key insight: the midpoint of the point and its image lies on the mirror line.

Across a general line Ax+By+C=0: the image of (x0,y0) is given by x'=x0-2A$\frac{Ax0+By0+C}{(A^2+B^2)}$, y'=y0-2B$\frac{Ax0+By0+C}{(A^2+B^2)}$. This is derived from two conditions: (1) the midpoint lies on the line, (2) the segment from point to image is perpendicular to the line.

The foot of perpendicular from (x0,y0) to Ax+By+C=0 is the midpoint of the point and its image: F=$\frac{(x0+x'}{2}$, $\frac{y0+y'}{2}$).

Reflecting a curve: to find the equation of the reflection of f(x,y)=0 across a line, substitute the inverse reflection (which is the same reflection formula, since reflection is its own inverse) into f.

JEE tip: always verify your reflection by checking that (1) the midpoint lies on the mirror line and (2) the segment is perpendicular to the mirror line.