A circle is a closed curve where every point on the curve is equidistant from a fixed point called the center. The problem at hand involves determining whether a given point is located inside a circle or not. Given a circle (coordinates of center and radius) and a point (coordinate), find if the point lies inside or on the circle, or not using JavaScript. 

Examples: 

Input: x = 4, y = 4 
       circle_x = 1, circle_y = 1, rad = 6; // Circle
Output: Inside 

Input: x = 3, y = 3 
circle_x = 0, circle_y = 1, rad = 2; // Circle
Output: Outside

These are the following approaches:

Using Distance Formula

One straightforward method to determine if a point lies inside a circle is by using the distance formula. The distance between the center of the circle (x1, y1) and the given point (x2, y2) is calculated using the formula:

distance = √((x2 - x1)² + (y2 - y1)²)

Note: If the calculated distance is less than the radius of the circle, i.e. point lies inside the circle.

Example: Implementation to find if a point lies inside a Circle using Distance Formula.

function pointInsideCircle(x, y, cx, cy, radius) {
    const distance =
        Math.sqrt((x - cx) ** 2 + (y - cy) ** 2);
    return distance < radius;
}
// Given Coordinates 
const x = 2, y = 3;
// Coordinates of the center of the circle
const cx = 0, cy = 0;
// Radius of the circle
const radius = 5;

console.log(pointInsideCircle(x, y, cx, cy, radius));

true

Time complexity: O(1)

Space complexity: O(1)

Using Circle Equation Approach

Another approach involves using the equation of a circle. The equation of a circle with center (h, k) and radius r is:

(x - h)² + (y - k)² = r²

If a point (x, y) satisfies this equation, it lies on the circle. To determine if the point lies inside the circle, we just need to check if the distance from the center (h, k) to (x, y) is less than the radius r.

Example: Implementation to find if a point lies inside a Circle using Circle Equation Approach.

function pointInsideCircle(x, y, cx, cy, radius) {
    const distanceSquared = 
        (x - cx) ** 2 + (y - cy) ** 2;
    return distanceSquared < radius ** 2;
}
// Given Coordinates 
const x = 2, y = 3;
// Coordinates of the center of the circle
const cx = 0, cy = 0;
// Radius of the circle
const radius = 5;

console.log(pointInsideCircle(x, y, cx, cy, radius)); 

true

Time complexity: O(1)

Space complexity: O(1)