To find the distance from a point to the center of a circle, we first need to identify the center of the circle. The standard equation of a circle is (x-h)²+(y-k)²=r², where (h, k) is the center of the circle and r is the radius. Given our equation (x+1)²+(y-4)²=64, we can immediately identify the center of our circle as (-1, 4).
The distance between two points in a plane can be calculated using the Distance Formula, which is derived from the Pythagorean Theorem. The distance d between two points (x1, y1) and (x2, y2) is given by:
d = √[(x2 - x1)² + (y2 - y1)²]
Our two points in this case are the center of the circle (-1, 4), and the given point (-1, -3).
Substituting (-1, 4) into (x1, y1) and (-1, -3) into (x2, y2) and applying these values in the Distance Formula, we get:
d = √[(-1--1)² + (-3-4)²]
= √[(0)² + (-7)²]
= √[0 + 49]
= √49
= 7
So, the distance from the point (-1,-3) to the center of the circle is 7 units.