Locus of point from which tangents drawn to a circle have a constant angle between them (Angle not equal to zero)

What to find : Locus of point from which tangents drawn to a circle have a particular angle (Angle not equal to zero) between them. 

Hello, Everyone. Before going to find our required solution, let us try to observe a phenomenon in the life.

  Figure : Illustration of variation of the angle with perpendicular distance from the wall

Assumptions : 

Let us assume that a person X is standing in front of a wall shown in above figure. Assume height of the wall is more than the person. Let A, B and C be the different positions of eye of the person as the person moves on the horizontal line segment shown in the above figure. Let "O" be the point on the wall lying on the horizontal line segment drawn from the eye to the wall such that it is perpendicular to the wall. Let "E" be the one end of the wall as shown in the above figure. In the above figure, variation of the angle means variation of the angle between the horizontal line segment and line segment joining the eye and point "E".

Observation : 
From the figure, Angle OAE > Angle OBE and Angle OBE >  Angle OCE. From this, we can say that, if the person X moves close to the wall from B then the angle (Angle between the horizontal line segment and line segment joining the eye and point "E") increases and if the person X moves away from the wall from B then the angle decreases. The movement is along the horizontal line segment shown in the above figure.

Conclusion : 
So, if the person X want to maintain the angle as constant, then the person should be at a constant perpendicular distance from the wall or else the angle may increase or decrease according to the movement.

Actual part of the question :
Using the above conclusion, let us try to solve this problem. We want locus of point from which tangents drawn to a circle have a particular angle between them (Angle not equal to zero). 

Figure : Illustration of variation of angle between the tangents with distance of the point from the center of the circle

Assumptions : 

Let us consider a circle as shown in above figure with center "C". Let the circle have its center at the origin. Let the horizontal line passing through the point "C" as shown in the above figure be X-axis. Let us consider points "P" and "Q" on the X-axis as shown in the above figure where Q is another position of the point "P" as it moves along X-axis. Tangents are drawn from the points P and Q to the circle as shown in the above figure. 

Note : 

The angle between the tangents is the double of the angle between a tangent and the X-axis since the point "P" is on the X-axis and the circle is symmetric along the X-axis. It implies that the angle between the tangents is depending on the angle between a tangent and the X-axis.

Observation : 

Similar to the previously observed phenomenon, the angle between the tangent and the X-axis depends on the distance of the point "P" from the circle along the X-axis.

Reason for the observation : 

We can draw only one tangent at a point on the circle and that tangent will meet the X-axis only at a single point. When the point "P" is moved on the X-axis, it implies that the point on the X-axis is changing due to which the tangent will change since a tangent at a point on the circle will meet the X-axis only at a single point.

Because of this, the angle between the tangent and the X-axis depends on the position of the point "P" which implies it depends on the distance of the point "P" from the circle along the X-axis.   

Explanation : 

So, to have a particular angle between tangents, we have to place the point "P" on the X-axis accordingly at a certain distance from the circle along the X-axis.

If we move the point from the position of required distance, the angle between the tangents may increase or decrease as the angle between the X-axis and a tangent may increase or decrease as shown in the above figure similar to the phenomenon shown previously. Therefore, the point should be at an appropriate distance from the circle to have a particular angle between the tangents. This statement can be changed as "The point should be at an appropriate distance from the center of the circle to have particular angle between the tangents" as we can measure the position of the point from the center of the circle also.

Then, if we remove the point from the X-axis and place it somewhere and try to observe the case to have the same angle between the tangents, it turns out to be same as the before because of the symmetry of the circle. i.e. The point should be at an appropriate distance from the center of the circle to have particular angle between the tangents.

Therefore, for a particular angle between the tangents, the point should be at a particular distance from the center of the circle. Therefore, collection of all such points will be a circle with the previous circle center as its center and the appropriate distance of the point from the center for particular angle between the tangents as its radius, according to the definition of circle.

Conclusion : 

Locus of point from which tangents drawn to a circle have constant particular angle between them is a circle with previous circle center as its center and the appropriate distance of the point from the center required for particular angle between the tangents as its radius.