Point of Tangency Calculator

Formula

To calculate the point of tangency on a circle:

\[ (x, y) = \left( x1 + \frac{r \times (y2 - y1)}{d}, y1 + \frac{r \times (x1 - x2)}{d} \right) \]

Where:

\((x, y)\) is the Point of Tangency
\((x1, y1)\) is the Center of the Circle
\((x2, y2)\) is the Point from which the Tangent Line is Drawn
\(r\) is the Radius of the Circle
\(d\) is the Distance between the Center of the Circle and the Point from which the Tangent Line is Drawn

What is a Point of Tangency?

A point of tangency is the exact spot where a line or curve (the tangent) touches another curve or surface without crossing it. In other words, it is the point where the tangent intersects the curve or surface. This point is significant in various fields such as mathematics, physics, and engineering, as it is often used in calculations and analyses related to curves and surfaces.

Example Calculation

Let's assume the following values:

Center X Coordinate (\(x1\)) = 0
Center Y Coordinate (\(y1\)) = 0
Tangent Point X Coordinate (\(x2\)) = 3
Tangent Point Y Coordinate (\(y2\)) = 4
Radius (\(r\)) = 5

Using the formula:

\[ d = \sqrt{(3 - 0)^2 + (4 - 0)^2} = \sqrt{9 + 16} = 5 \]

\[ (x, y) = \left( 0 + \frac{5 \times (4 - 0)}{5}, 0 + \frac{5 \times (0 - 3)}{5} \right) = (4, -3) \]

The point of tangency is (4, -3).