If there are two intersection points, they are the \((x, y)\) coordinates of the points where the circle and the line intersect. If there is only one intersection point, it is where the line is tangent to the circle. If there are no intersection points, it means the circle and the line do not intersect in the given coordinate system.
Circle Parameters
| Parameter | Value |
|---|---|
| Center X (h) | |
| Center Y (k) | |
| Radius (r) |
Straight Line Parameters
| Parameter | Value |
|---|---|
| Slope (m) | |
| Y-Intercept (b) |
Intersection Point
Standard Equation of a Circle
The standard form of the equation of a circle in a Cartesian coordinate system is:
\[
(x – h)^2 + (y – k)^2 = r^2
\]
In this equation:
- \((h, k)\) represents the coordinates of the center of the circle.
- \(r\) represents the radius of the circle.
This equation describes all the points \((x, y)\) that are equidistant from the center \((h, k)\) by a distance of \(r\). In other words, it defines a circle with center \((h, k)\) and radius \(r\) in the coordinate plane.
Standard Equation of a Straight Line
The standard form of the equation of a straight line in a Cartesian coordinate system is typically written as:
\[
Ax + By = C
\]
In this equation:
- \(A\), \(B\), and \(C\) are constants.
- \(A\) and \(B\) are not both equal to 0 (to avoid degenerate cases).
This form is often preferred for equations of lines because it allows for a straightforward representation of a linear relationship between \(x\) and \(y\). The values of \(A\), \(B\), and \(C\) can be determined from various representations of a line, such as the slope-intercept form (\(y = mx + b\)) or the point-slope form (\(y – y_1 = m(x – x_1)\)), where \(m\) is the slope of the line, and \((x_1, y_1)\) is a point on the line.
How to find the Intersection Points of a Circle and a Straight Line?
To find the intersection points of a circle and a straight line, you need to solve their equations simultaneously. The circle is represented by its equation in standard form:
\[
(x – h)^2 + (y – k)^2 = r^2
\]
And the straight line is represented by its equation in standard form:
\[
Ax + By = C
\]
Steps Involved:
- Write the equations of the circle and the straight line.
- Substitute the expression for \(y\) from the line equation into the circle equation.
- Expand and simplify the equation.
- Solve for \(x\). This may involve expanding the equation, rearranging terms, and applying algebraic techniques to isolate \(x\). You may have to solve a quadratic equation, so it might lead to zero, one, or two possible values for \(x\).
- Once you have the values of \(x\), substitute them back into the line equation to find the corresponding \(y\)-coordinates.
- These \(x\) and \(y\) coordinates represent the intersection points of the circle and the straight line. Depending on the nature of the circle and the line, there may be zero, one, or two intersection points.