In geometry, tangents and normals have great importance. A straight line that touches a circle at only one point is the tangent of the circle. The normal to a circle is a straight line drawn at 90∘ to the tangent at the point where the tangent touches the circle. In this article, we will discuss how to find the tangent and normal to a circle.
Step 1. If x2 + y2 = a2 is a circle, then
a. The equation of a tangent to the circle at (x1, y1) is given by xx1 + yy1 = a2.
b. The equation of normal to the circle at (x1, y1) is given by yx1 – xy1 = 0.
c. The equation of a tangent to the circle at (a cos θ, a sin θ) is given by x cos θ + y sin θ = a.
d. The equation of a normal to the circle at (a cos θ, a sin θ) is given by x sin θ – y cos θ = 0.
Step 2. If the circle is given by x2 + y2 + 2gx + 2fy + c = 0
a. The equation of a tangent to the circle at (x1, y1) is xx1 + yy1 + g(x + x1) + f(y + y1) + c = 0
b. The equation of normal to the circle at (x1, y1) is (y – y1)/(y1 + f) = (x – x1)/(x1 + g).
Step 3. For a line y = mx + c to be a tangent to a circle x2 + y2 = a2, it should satisfy c = ± a(√(1+m2). The equation of tangent is given by y = mx ± a(√(1+m2).
Particular Cases of Circle
Example 1: The tangent to circle
Equation of tangent to
Thus, point of contact is (3, -1).
Example 2: Find the angle between the two tangents from the origin to the circle
Any line through (0, 0) be
The product of both the slopes is -1.
Hence, the angle between the two tangents is
Example 3: Find the equation of normal to the circle
The centre of the circle is (1/2, 5/4)
Normal to circle at point (1, 1) is line passing through the points (1, 1) and (1/2, 5/4) which is