Given that we need to find the equation of the circle which is concentric with x² + y² - 6x + 12y + 15 = 0 and double its area.

We know that concentric circles will have the same centre.

Let us assume the concentric circle be x² + y² - 6x + 12y + c = 0. .....(ii)

We know that for a circle x² + y² + 2ax + 2by + c = 0 - - (1)

⇒ Centre = (- a, - b)

⇒ Radius =

Let us assume the radius of the first circle is r₁.

⇒

⇒

⇒

Given that the concentric circle’s area is double the first circle’s area.

Let us assume the radius of the concentric circle be r₂.

We know that the area of the circle is πr²

Now,

⇒ Area of concentric circle = 2 × area of the first circle

⇒

⇒

⇒ r₂² = 60

⇒ r₂ = √60

Comparing with (ii) with (1), we get

⇒

⇒ 9 + 36 - c = 60

⇒ c = - 15

Substituting c value in (ii), we get

⇒ x² + y² - 6x + 12y - 15 = 0

∴ The equation of the concentric circle is x² + y² - 6x + 12y - 15 = 0.