line with equation intersect the circle with radius 6 and center (0, 3)

The formula for the equation of a circle with center=(h,k) and radius = r is:

     (x-h)²+(y-k)² =r³

For this problem h = 0, k = 3 and r = 6:

     x² + (y-3)² = 6²

Since y = 1.5x + 3, to find the points of intersection, replace y in the circle equation with this value:

     x² + (1.5x + 3 - 3)² = 6²

    --->      x² + (1.5x)² = 36           (simplify)

     --->    x² + 2.25x² = 36 

     --->     3.25x² = 36

     --->     x² = 36 / 3.25

     --->     x = ±√(36 / 3.25)

Now, find the corresponding value for y for each of these values for x and you will have the two points of intersection.