Given that we need to find the equation of the incircle formed by the coordinate axes and the line 4x + 3y = 6.

Let us find the vertices of triangle.

We know that the axes meet at origin O.

When the line meets x - axis, the value of ‘y’ is 0.

⇒ 4x + 3(0) = 6

⇒

⇒

The point is A.

When the line meets y - axis, the value of ‘x’ is 0.

⇒ 4(0) + 3y = 6

⇒

⇒ y = 2

The point is B(0,2).

Let us find the length of sides of the triangle.

We know that the distance between two points (x₁,y₁) and (x₂,y₂) is .

⇒

⇒

⇒

⇒

⇒

⇒ OB = b = 2

⇒

⇒

⇒

⇒

We know that incentre of a triangle is given by:

⇒

⇒

⇒

⇒

We have x = 0 as tangent to this circle.

We know that the perpendicular distance between the circle and centre is equal to the radius of the circle.

We know that the perpendicular distance between from the point (x₁,y₁) to the line ax + by + c = 0 is

⇒

⇒ .

We know that the equation of the circle with centre (p, q) and having radius ‘r’ is given by:

⇒ (x - p)² + (y - q)² = r²

Now we substitute the corresponding values in the equation:

⇒

⇒

⇒ 4(x² + y² - x - y) + 1 = 0

∴The correct option is (b).