Question

Find the coordinates of the centre and radius of the following circle:

1/2 (x² + y²) + x cos θ + y sin θ − 4 = 0

Answer 1

Given:

The equation of the circle is,

1/2 (x² + y²) + x cos θ + y sin θ − 4 = 0

(Multiply by 2 we get)

x² + y² + 2x cos θ + 2y sin θ – 8 = 0

By comparing with the equation x² + y² + 2ax + 2by + c = 0

Centre = (−a, −b)

= [(−2 cos θ)/2, (−2 sin θ)/2]

= (−cos θ, −sin θ)

Radius = √(a² + b² − c)

= √[(−cos θ)² + (sin θ)² −(−8)]

= √[cos²θ + sin²θ + 8]

= √[1 + 8]

= √[9]

= 3

∴ The centre and radius of the circle is (−cos θ, −sin θ) and 3.