If centre of the circle lies on the intersection of lines x + y = 1 and y = 2 and also cuts the circle x² + y² = 9 orthogonally then find the equation of the circle.

Concept:

• Condition of orthogonality = 2g₁g₂ + 2f₁f₂ = c₁ + c₂

Calculation:

Centre lies on intersection of lines x + y = 1 and y = 2,

So point of intersection = (-1, 2) = (-g, -f)

So g = 1, f = -2

Given circle = x² + y² - 9 = 0

Lets equation of that orthogonal circle is x² + y² + 2gx + 2fy + c = 0

Checking for the condition 2g₁g₂ + 2f₁f₂ = c₁ + c₂

⇒ 2. g. (0) + 2. f. (0) = c - 9

⇒ c = 9

So g = 1, f = -2 and c = 9

Radius of circle = \(\sqrt {{g^2} + {f^2} - c} \) = \(\sqrt {{1^2} + {{\left( { - 2} \right)}^2} - 9} \)