For AI agents: use /llms.txt for the Nakafa content index.
A circle tangent line is a line that intersects the circle at exactly one point. The intersection point between the tangent line and the circle is called the point of tangency.
Important property: A tangent line is always perpendicular to the radius of the circle at the point of tangency.
If point lies on the circle , then the equation of the tangent line at that point is:
For a circle with center :
The equation of the tangent line to circle with gradient is:
For a circle with center :
From a point outside the circle, two tangent lines can be drawn to the circle.
If is a point outside the circle with center and radius , then the length of the tangent line from P to the circle is:
External common tangent lines are lines that touch both circles and do not intersect the line connecting the two circle centers.
Length of external common tangent line:
where is the distance between the two circle centers.
Internal common tangent lines are lines that touch both circles and intersect the line connecting the two circle centers.
Length of internal common tangent line:
Find the equation of the tangent line to circle at point .
Solution:
Since point lies on the circle (can be verified: ), the equation of the tangent line is:
Find the equation of the tangent line to circle that is parallel to line .
Solution:
The gradient of line is .
Equation of tangent line with gradient :
Therefore, the equations of the tangent lines are:
Find the length of the tangent line from point to circle .
Solution:
Circle center and radius .
Find the equation of the tangent line to circle at point !
Find the equation of the tangent line to circle that is perpendicular to line !
From point tangent lines are drawn to circle . Find:
Two circles are centered at with radius and with radius . Find the length of the external common tangent line!
Find the equation of the tangent line to circle that passes through point !
Tangent line equation at a point on the circle
Verify point on circle:
Tangent line equation:
Tangent line perpendicular to a given line
Gradient of line is .
Since perpendicular, then .
Tangent line from external point
Length of tangent line:
External common tangent line
Tangent line through external point
Circle:
Center , radius
Tangent line equation:
Therefore:
Coordinates of tangent points can be found using the tangent line equation from external point.
Verify point outside circle:
Point is exactly on the circle! Therefore the tangent line equation is:
