Touches the y-axis at the point (0,3) suggests that it is a tangent to the circle.

Since tangent is perpendicular to radius, and this suggests that a horizontal line y = 3 cuts through the centre since the tangent point (0,3) lies on both the vertical tangent line and the radius. i.e the y-coordinate in this line is 3.

So the centre has y-coordinate of 3.

One of its intercepts with the x-axis is 9 means the coordinates of that point are (9,0)

Since the y-axis is a tangent and the one if the intercept with the x-axis is positive, it means the other intercept is also positive. So the circle lies on the positive x-axis.

Use the standard form for equation of circle (x - a)² + (y - b)² = r²

Sub (0,3) , b = 3,
(0 - a)² + (3 - 3)² = r²
a² = r²

Sub (9,0), b = 3, a² = r²
(9 - a)² + (0 - 3)² = a²
81 - 18a + a² + 9 = a²
90 = 18a
a = 5

Therefore r² = 5² = 25

So the centre is (5,3) and the equation is :
(x - 5)² + (y - 3)² = 25

O thank you for explaining to me :)

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