# Intersection of a Line and a Circle

## Intersection of a Line and a Circle:

Let us consider the line y = mx + c and the circle x2 + y2 = r2 (whose center is at the origin). For the point of intersection of the line and the circle, we have

(i) If c2 < r2(1 + m2), then D > 0. The roots x1 and x2 are then real and distinct. In this case, the line y = mx + c intersects the circle x2 + y2 = r2 at two distinct points A and B as shown in the figure below. Here, r > | c / √(1 + m2) | i.e., the radius of the circle is more than the distance of the line from the center.

(ii) If c2 = r2(1 + m2), then D = 0. The roots x1 and x2 of equation (i) are equal. The line y = mx + c, in this case, intersects the circle x2 + y2 = a2 at two real and coincident points as shown in the figure below. Here, r = | c / √(1 + m2) i.e., the radius of the circle is equal to the distance of the line from the center.

(iii) If c2 > r2(1 + m2), then D < 0. The roots x1 and x2 of equation (i) are imaginary. The line y = mx + c does not intersect the circle as shown in the figure below. In this case, r < | c / √(1 + m2) | i.e., the radius of the circle is less than the distance of the line from the center of the circle.

## Length of the Intercepted Chord:

Let the line y = mx + c intersect the circle at two points A and B as shown in the figure. If the coordinates of points A and B be (x1, y1) and (x2, y2) respectively. The length of the chord is AB.

Note: