Calculating the Shortest Distance Between a Point and a Circle

Determining the shortest distance between a point and a circle is an essential problem in geometry and has applications in various fields, including computer graphics, robotics, and engineering. In this article, we will explore the method to find the smallest distance between the point (-2, -2) and a point on the circumference of the circle given by the equation (x - 1)^2 (y - 2)^2 4.

The Equation of the Circle

First, let's identify the center and radius of the circle based on the given equation:

The standard form of the equation of a circle is (x - h)2 (y - k)2 r2, where (h, k) is the center of the circle and r is the radius.

Given the equation (x - 1)2 (y - 2)2 4, we can see that the center of the circle is at (1, 2), and the radius is 2 since 4 22.

Calculating the Distance to the Center

Next, we need to calculate the distance from the point (-2, -2) to the center of the circle (1, 2) using the distance formula:

d √((x? - x?)2 (y? - y?)2)

Substituting the coordinates, we get:

d √((1 - (-2))2 (2 - (-2))2) √((1 2)2 (2 2)2) √(32 42) √(9 16) √25 5

The distance from the point (-2, -2) to the center of the circle (1, 2) is 5 units.

Shortest Distance to the Circumference

The shortest distance from a point to the circumference of a circle is the distance from the point to the center of the circle minus the radius of the circle. In this case, the shortest distance is:

Shortest distance 5 - 2 3

Therefore, the smallest distance between the point (-2, -2) and a point on the circumference of the circle is 3 units.

Geometric Visualization

The equation (x - 1)2 (y - 2)2 4 represents a circle centered at (1, 2) with a radius of 2. The point (-2, -2) is located in the bottom-left quadrant of the coordinate plane. The shortest distance between this point and the circle is a straight line (a tangent) that is perpendicular to a line connecting (-2, -2) and the center (1, 2).

Solving for the distance geometrically, we can use the distance formula and then subtract the radius:

Distance to the center √((-2 - 1)2 (-2 - 2)2) √((-3)2 (-4)2) √(9 16) √25 5

Radius of the circle √4 2

Shortest distance to the circle 5 - 2 3

Since in Euclidean space, the straight line is the shortest path, this is the shortest distance between the point and the circle.