The equation of the circle drawn with the focus of the parabola (x – 1 )2 – 8y  = 0 as its centre and touching the parabola at its vertex is :

To find the equation of the circle, we need to determine its center and radius.

Step 1: Find the vertex of the parabola. The given equation is in the form (x - h)^2 = 4a(y - k), where (h, k) represents the vertex. Comparing this with the given equation (x - 1)^2 - 8y = 0, we can see that the vertex is (1, 0).

Step 2: Find the radius of the circle. The radius of the circle is the distance between the vertex of the parabola and the point where the circle touches the parabola. Since the circle touches the parabola at its vertex, the radius is the distance between the vertex and the focus of the parabola.

Step 3: Find the focus of the parabola. The focus of the parabola is given by the equation (h, k + 1/4a), where a is the coefficient of y in the equation of the parabola. In this case, a = -8, so the focus is (1, 0 + 1/4(-8)) = (1, -1).

Step 4: Find the radius. The radius is the distance between the vertex (1, 0) and the focus (1, -1). Using the distance formula, we have: radius = sqrt((1 - 1)^2 + (0 - (-1))^2) = sqrt(0 + 1) = 1.

Step 5: Write the equation of the circle. The equation of a circle with center (h, k) and radius r is given by (x - h)^2 + (y - k)^2 = r^2. Substituting the values we found, we have: (x - 1)^2 + (y - 0)^2 = 1^2.

Therefore, the equation of the circle drawn with the focus of the parabola (x - 1)^2 - 8y = 0 as its center and touching the parabola at its vertex is (x - 1)^2 + y^2 = 1.