How to Find Tangent and Normal to a Circle - Testbook

Procedure to Determine Tangent and Normal to a Circle

Step 1: For a circle defined by the equation x² + y² = a², the following hold true:

a. The equation of the tangent to the circle at a point (x₁, y₁) is given by xx₁ + yy₁ = a².

b. The equation of the normal at the point (x₁, y₁) is yx₁ – xy₁ = 0.

c. The equation for a tangent to the circle at (a cos θ, a sin θ) is x cos θ + y sin θ = a.

d. The equation for a normal to the circle at (a cos θ, a sin θ) is x sin θ – y cos θ = 0.

Step 2: If the circle's equation is x² + y² + 2gx + 2fy + c = 0:

a. The equation of the tangent at the point (x₁, y₁) is xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0.

b. The equation of the normal at the point (x₁, y₁) is (y – y₁)/(y₁ + f) = (x – x₁)/(x₁ + g).

Step 3: For a line y = mx + c to be a tangent to the circle x² + y² = a², the condition c = ± a(√(1+m²) must be satisfied. The tangent's equation is therefore given by y = mx ± a(√(1+m²).

Sample Problems:

Example 1: The tangent to the circle x² + y² = 9 at the point (2, -3) also touches the circle x² + y² – 6x + 4y + 16 = 0. Determine the coordinates of the corresponding point of contact.

Example 2: Calculate the angle between the two tangents drawn from the origin to the circle (x – 5)² + (y + 2)² = 16.

Example 3: Determine the equation of the normal to the circle 2x² + 2y² – 2x – 4y + 2 = 0 at the point (2, 2).