minase said:I'm trying to find the points of intersection
of line and circle with equations:
(x-p)^2 + (y-q)^2 = r^2
(y-y1)*(x2-x1)-(x-x1)*(y2-y1)=0
but i can't handle with this. Can anyone help me?
mathman said:Your second equation (the line) can be simplified to look like y=mx+b (I assume x1, x2,y1,y2 are constants). Substitute mx+b for y in the first equation. You now have a quadratic in x. Solve for x, 2 real roots gives points of intersection, 1 root is tangency point, 0 real roots means no intersection. If x roots are real, use second equation to get y values.
Special case x1=x2, then x comes right out of second equation and 2 values of y can be found. Above comments about real roots and intersections apply. Line happens to be vertical.
To find the intersection points of a line and a circle, you can use the equation of the line and the equation of the circle. Substitute the y value from the line equation into the circle equation and solve for the x values. Then, substitute the x values into the line equation to find the corresponding y values. These points represent the intersections between the line and the circle.
No, a line can only intersect a circle at a maximum of two points. This is because a line is a straight path that extends infinitely in both directions, while a circle is a closed shape with a finite circumference. Therefore, a line can only intersect a circle at two points at most.
When a line and a circle do not intersect, it means that there are no points that lie on both the line and the circle. This could happen if the line and circle are parallel and do not touch at any point, or if the line is outside of the circle and does not cross its circumference.
A line is tangent to a circle if it touches the circle at exactly one point. This means that the line is perpendicular to the radius of the circle at the point of intersection. To determine if a line is tangent to a circle, you can find the equation of the line and the equation of the circle and set them equal to each other. If the resulting equation has only one solution, then the line is tangent to the circle.
Yes, it is possible for a line and a circle to have no intersection points. This can happen if the line is outside of the circle and does not cross its circumference, or if the line is parallel to the circle and does not touch it at any point. In these cases, there are no points that lie on both the line and the circle, so there are no intersection points.