Distance Of A Point From A Line

To find the equation of the locus of such that the line segment joining points and subtends an angle of , we can use the concept of circle power and angle subtension.

The general concept is that if a line segment subtends angle at point , then the point lies on a circle known as the circle of Apollonius with as a chord for given angle .

Given points are and . Using the formula for angle subtended by a chord at any point, we know that if , then point lies on a combination of two lines or a circle through a specific configuration.

The condition for angle at is given by the equation:

Here , hence: (the product of slopes must be -1 for perpendicularity, which is directly related to ).

The locus of point is given by the equation of the radical axis which is derived from two circles:

- The first circle is centered at with arbitrary radius: .

- The second circle is centered at with same radius: .

The difference between these two equations gives us the radical axis:

Considering the path of the point on the plane making :

The circle centers, which are points of equal scaled distance from and , give:
and

This results in two possibilities where either circle condition holds:

Thus the correct locus equation is:

Additionally, as they would invalidate the range of possible lines considered.