The area (in sq. units) of the circle touching the line x + y = 4 at 1 , 3 and intersecting x 2 + y 2 = 4 orthogonally is equal to

Step-by-step solution

Let equation of the circle is x - 1 2 + y - 3 2 + α x + y - 4 = 0 ⇒ x 2 + y 2 + α - 2 x + α - 6 y + 10 - 4 α = 0 Which is orthogonal to x 2 + y 2 = 4 Now, 2 g 1 g 2 + 2 f 1 f 2 = c 1 + c 2 ⇒ 0 + 0 = 10 - 4 α - 4 ⇒ α = 3 2 ⇒ Equation of the circle is x 2 + y 2 - x 2 - 9 y 2 + 4 = 0 Radius = 1 16 + 81 16 - 4 = 18 16 = 3 2 2 units ⇒ area of the circle is 9 π 8 sq. units