Question

Show that the line 3x − 4y − 1 = 0 touches the circle x² + y² − 2x + 4y + 1 = 0 and find the coordinates of the point of contact.

Answer 1

The centre of the circle is (1, −2) and its radius is √1² + 2² - 1 = 2. The distance of the line from the centre (1, −2) is given by

which is equal to the radius of the circle. Therefore, the line touches the circle. Let (x₁, y₁) be the point of contact. That is, at (x₁, y₁), the line

3x - 4y - 1 = 0  ....(1)

is the tangent. But, by Theorem 3.7, the equation of the tangent at (x₁, y₁) is

S₁ ≡ xx₁ + yy₁ - (x + x₁)  + 2(y + y₁) + 1 = 0

That is,

S₁ ≡ (x₁ - 1) x + (y₁ + 2)y - x₁ + 2y₁ + 1 = 0   ......(2)

Equations (1) and (2) represent the same straight line. Therefore,

Hence

(x₁,y₁) = (-1/5,-2/5)