Tangent Equation of Circle & Point of Contact

Intersection of straight line and a circle: Let the equation of circle be x² + y² = a² and the equation of the line be y = mx + c then

⇒ When points of intersection are real and distinct: then length of perpendicular from centre should be less the radius.

⇒ When points of intersection are coincident: line touches the circle if the length perpendicular from centre is equal to radius.

⇒ When points of intersection are imaginary: Line does not intersect a circle if the length of perpendicular from centre is greater than radius of circle.

Length of the intercept cut off from the line y = mx + c by the circle x² + y² = a² is

Condition of Tangency: The line y = mx + c touches the circle x² + y² = a² if the length of the intercepts is zero i.e., c = ± a √(1 + m²)

Tangent to a circle: Let P be a point on circle and let PQ be secant

Tangent at point P is the limiting position of a secant PQ when Q tends to P along the circle. The point P is called the point of contact of the tangent.

Different forms of equation of tangents:

• The equation of tangent of slope m to the circle x² + y² = a² is y = mx ± a √(1 + m²) the coordinates of the point of contact are 
• The equations of tangents of slope m to the circle (x - a)² + (y - b)² = r² are given by y – b = m (x - a) ± r √(1 + m²) and the coordinates of point of contact are 
• The equations of the tangents of slope m to circle x² + y² + 2gx + 2fy + c = 0 is y + f = m(x + g) ± 

Point form: The equation of the tangent at the point P(x₁, y₁) to a circle x² + y² + 2gx + 2fy + c = 0 is xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0

Parametric form: The equation of the tangent to the circle x² + y² = a² at the point (a cosθ, a sinθ) is x cosθ + y sinθ = a.

The equation of tangent to the circle (x - a)² + (y - b)² = r² at the point (a + r cosθ, b + r sinθ) is (x - a) cosθ + (y - b) sinθ = r.