Is a line ax+by+c=0 tangent to a circle?

To determine whether the line represented by the equation ax + by + c = 0 is tangent to a circle, one must analyse the geometric relationship between the line and the circle. A line is considered tangent to a circle if it intersects the circle at exactly one point. This condition can be mathematically expressed by examining the distance from the centre of the circle to the line and comparing it to the radius of the circle.

The equation of a circle can be expressed in the standard form (x – h)² + (y – k)² = r², where (h, k) denotes the centre of the circle and r represents its radius. To find the distance from the centre of the circle to the line, one can utilise the formula for the distance from a point to a line. Specifically, the distance D from the point (h, k) to the line ax + by + c = 0 is given by the formula D = |ah + bk + c| / √(a² + b²). For the line to be tangent to the circle, this distance must equal the radius r.

If the calculated distance D is equal to the radius r, it confirms that the line is tangent to the circle, intersecting it at precisely one point. Conversely, if D is less than r, the line intersects the circle at two points, and if D is greater than r, the line does not intersect the circle at all. Thus, the relationship between the line and the circle can be conclusively established through this analysis of distances and geometric properties.