How to Find Tangent and Normal to a Circle

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. A normal will pass through the centre of the circle. Tangents and normals to a circle is an important topic. In this article, we will discuss how to find the tangent and normal to a circle.

Steps to find 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).

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