Length of Intercept Made by a Circle on a Line

The length of the intercept made by a circle on a line refers to the segment of the line that lies inside the circle. This segment is essentially the chord of the circle that the line cuts through. To understand this concept, we need to delve into the geometry of circles and the equations that represent them.

Circle Equation

A circle with center at point $(h, k)$ and radius $r$ can be described by the equation:

$$(x - h)^2 + (y - k)^2 = r^2$$

Line Equation

A line in the plane can be represented in various forms, one of which is the slope-intercept form:

$$y = mx + c$$

where $m$ is the slope of the line and $c$ is the y-intercept.

Intersection of Circle and Line

To find the length of the intercept, we need to determine the points where the line intersects the circle. This is done by solving the system of equations formed by the circle's equation and the line's equation.

Calculation of Intercept Length

The length of the intercept can be found by using the following steps:

1. Substitute the line equation into the circle equation to find the points of intersection.
2. Solve the resulting quadratic equation to find the x-coordinates of the intersection points.
3. Substitute the x-coordinates back into the line equation to find the corresponding y-coordinates.
4. Use the distance formula to calculate the distance between the two points of intersection, which gives the length of the intercept.

The distance formula is given by:

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of the two intersection points.

Example

Let's consider a circle with center at $(0, 0)$ and radius $5$ units, and a line with the equation $y = 2x + 1$.

1. The circle's equation is $x^2 + y^2 = 25$.
2. Substituting the line equation into the circle's equation gives us $x^2 + (2x + 1)^2 = 25$.
3. Expanding and simplifying, we get a quadratic equation: $5x^2 + 4x - 24 = 0$.
4. Solving this quadratic equation, we find the x-coordinates of the intersection points: $x_1$ and $x_2$.
5. We then find the corresponding y-coordinates by substituting $x_1$ and $x_2$ into the line equation.
6. Finally, we use the distance formula to calculate the length of the intercept.

Table of Differences and Important Points

Aspect	Circle	Line	Intercept Length
Basic Definition	A set of points equidistant from a center point.	An infinite set of points extending in two directions.	The segment of a line that lies inside a circle.
Equation	$(x - h)^2 + (y - k)^2 = r^2$	$y = mx + c$	Calculated using intersection points.
Parameters	Center $(h, k)$, Radius $r$	Slope $m$, Y-intercept $c$	Distance between two points.
Intersection Points	Found by solving system of equations.	Determined by where it crosses other geometric figures.	Found by substituting line equation into circle equation.
Calculation Method	Not applicable.	Not applicable.	Use distance formula after finding intersection points.

Conclusion

The length of the intercept made by a circle on a line is an important concept in geometry, particularly when dealing with the intersections of different shapes. By understanding the equations that represent circles and lines, and by following the steps outlined above, one can calculate the length of the intercept with precision. This knowledge is not only useful in academic settings but also in various practical applications such as engineering and design.