What is the Formula Derivation to find new coordinates when axes are rotated?

A line has intercepls a and b on the coordinate axes. When the axes are rotated an angle alpha Keeping the origin fixed. The line makes equal intercepts on the coordinate axes, then tan alpha =

What was the old coordinates of point A(2,5) if coordinate axes is rotated by 30^(@) in clockwise direction.

A point (1, 1) undergoes reflection in the x-axis and then the coordinates axes are rotated through an angle of pi/4 in anticlockwise direction. The final position of the point in the new coordinate system is

The line L has intercepts a and b on the coordinate axes. The coordinate axes are rotated through a fixed angle, keeping the origin fixed. If p and q are the intercepts of the line L on the new axes, then (1)/(a^(2))-(1)/(p^(2))+(1)/(b^(2))-(1)/(q^(2)) is equal to

If the coordinates of a point P changes to (2,-6) when the coordinate axes are rotated through an angle of 135^(@) , then the coordinates of P in the original system are

Write the new coordinates of the following point (0,5) when the axes are rotated through to an angle 30^0