Find the condition that the tangents drawn from 0 , 0 to S ≡ x 2 + y 2 + 2 g x + 2 f y + c = 0 be perpendicular to each other.

The equation of the given circle is S ≡ x 2 + y 2 + 2 g x + 2 f y + c = 0 . The centre of the circle S would be at - g , - f and the radius r = g 2 + f 2 - c . If the tangents are perpendicular to each other then the angle between them is π 2 . As we know that if θ is the angle between the tangents from a point x 1 , y 1 to a circle S = 0 then tan θ 2 = r S 11 So, tan θ 2 = g 2 + f 2 - c 0 + 0 + 2 g · 0 + 2 f · 0 + c = g 2 + f 2 - c c ⇒ tan π 4 = g 2 + f 2 - c c ⇒ 1 = g 2 + f 2 - c c ⇒ g 2 + f 2 - c = c Thus, the required condition would be g 2 + f 2 = 2 c .

Find the condition that the tangents drawn from 0 , 0 to S ≡ x 2 + y 2 + 2 g x + 2 f y + c = 0 be perpendicular to each other.

The equation of the given circle is S ≡ x 2 + y 2 + 2 g x + 2 f y + c = 0 . The centre of the circle S would be at - g , - f and the radius r = g 2 + f 2 - c . If the tangents are perpendicular to each other then the angle between them is π 2 . As we know that if θ is the angle between the tangents from a point x 1 , y 1 to a circle S = 0 then tan θ 2 = r S 11 So, tan θ 2 = g 2 + f 2 - c 0 + 0 + 2 g · 0 + 2 f · 0 + c = g 2 + f 2 - c c ⇒ tan π 4 = g 2 + f 2 - c c ⇒ 1 = g 2 + f 2 - c c ⇒ g 2 + f 2 - c = c Thus, the required condition would be g 2 + f 2 = 2 c .