Find the locus of the point of intersection of the tangent to any circle and the perpendicular let fall on this tangent from a fixed point on the circle.

Answer

To find the locus of the point of intersection of the tangent to any circle and the perpendicular dropped from a fixed point on the circle to this tangent, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Let the circle be centered at the origin \( O(0, 0) \) with radius \( A \). The equation of the circle is \( x^2 + y^2 = A^2 \).
- Let \( C \) be a point on the circle, which can be represented in polar coordinates as \( C(A \cos \theta, A \sin \theta) \).

2. **Finding the Tangent Line**:
- The equation of the tangent line at point \( C \) is given by:
\[ x \cos \theta + y \sin \theta = A \]

3. **Dropping a Perpendicular**:
- From point \( C \), drop a perpendicular to the tangent line. Let the foot of the perpendicular be point \( N \).

4. **Finding the Coordinates of Point \( N \)**:
- The distance from the center \( O \) to the tangent line can be calculated using the formula for the distance from a point to a line. The distance \( d \) from the origin \( O(0, 0) \) to the tangent line is:
\[ d = \frac{|0 \cdot \cos \theta + 0 \cdot \sin \theta - A|}{\sqrt{\cos^2 \theta + \sin^2 \theta}} = \frac{A}{1} = A \]
- Therefore, the coordinates of point \( N \) can be expressed as:
\[ N = (A \cos \theta, A \sin \theta - A) \]

5. **Finding the Point of Intersection \( P \)**:
- The point \( P \) is the intersection of the tangent line and the perpendicular from point \( C \). The coordinates of point \( P \) can be expressed in terms of \( R \) and \( \theta \):
\[ OP = ON + NP \]
- Here, \( ON = A \cos \theta \) and \( NP = A \).

6. **Using the Relationship**:
- From the triangle formed, we can express the distance \( OP \) as:
\[ OP = A \cos \theta + A \]
- Thus, we have:
\[ R = A(1 + \cos \theta) \]

7. **Identifying the Locus**:
- The equation \( R = A(1 + \cos \theta) \) represents a cardioid in polar coordinates.

### Conclusion:
The locus of the point of intersection of the tangent to any circle and the perpendicular dropped from a fixed point on the circle is given by the equation:
\[ R = A(1 + \cos \theta) \]
This is the equation of a cardioid.