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.
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).
Particular Cases of Circle
Solve Examples:
Example 1: The tangent to circle x2 + y2 = 5 at (1, -2) also touches the circle x2 + y2 – 8x + 6y + 20 = 0. Find the coordinates of the corresponding point of contact.
Solution:
Equation of tangent to x2 + y2 = 5 at (1, -2) is
or
putting x = 2y + 5 in second circle, we get
Thus, point of contact is (3, -1).
Example 2: Find the angle between the two tangents from the origin to the circle (x – 7)2 + (y + 1)2 = 25.
Solution:
Any line through (0, 0) be y – mx = 0 and it is a tangent to circle (x – 7)2 + (y + 1)2 = 25.
If
The product of both slopes is -1.
Hence, the angle between the two tangents is π/2.
Example 3: Find the equation of normal to the circle 2x2 + 2y2 – 2x – 5y + 3 = 0 at (1, 1).
Solution:
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 x + 2y = 3.
Frequently Asked Questions
What do you mean by a tangent to a circle?
A tangent is a straight line that touches a circle at only one point.
Give the equation of a tangent to the circle x2 + y2 = a2 at (x1, y1).
The equation of the tangent to the circle x2 + y2 = a2 at (x1, y1) is given by xx1 + yy1 = a2.
Give the condition for a line y = mx + c to be a tangent to a circle x2 + y2 = a2.
For a line y = mx + c to be a tangent to a circle x2 + y2 = a2 , it should satisfy the condition c = ± a(√(1+m2).
Give the equation of the normal to the circle x2 + y2 = a2 at (x1, y1).
The equation of normal to the circle x2 + y2 = a2 at (x1, y1) is given by yx1 – xy1 = 0.
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