Determine if a Point Lies Inside, Outside, or on a Circle: The Example of (4, 2) on x-1^2y-4^222

When dealing with circles in coordinate geometry, one often encounters the task of determining the position of a point relative to the circle. This involves checking if a point lies on, inside, or outside the circle. We will explore this process using the example of a circle with the equation (x - 1^2 y - 4^2 22) and a given point ((4, 2)).

Standard Equation of the Circle and Center

The given equation is (x - 1^2 y - 4^2 22). This is the standard form of a circle's equation, which can be rewritten as:

[(x - 1)^2 (y - 4)^2 22]

This equation represents a circle with center ((1, 4)) and radius (sqrt{22}).

Calculating the Distance from the Point to the Center

To determine if the point ((4, 2)) lies inside, outside, or on the circle, we need to measure the distance from the point to the center of the circle. The distance (d) between the points ((4, 2)) and ((1, 4)) is given by the distance formula:

[d sqrt{(4 - 1)^2 (2 - 4)^2} sqrt{3^2 (-2)^2} sqrt{9 4} sqrt{13}]

Now, we compare this distance with the radius of the circle. The radius is (sqrt{22}).

Comparison of Distance and Radius

Let's compare (sqrt{13}) and (sqrt{22}):

(sqrt{13}

Since the distance (d) from the point ((4, 2)) to the center ((1, 4)) is less than the radius (sqrt{22}), the point ((4, 2)) lies inside the circle.

Alternative Methods to Verify the Point's Position

Another way to verify if the point ((4, 2)) lies inside the circle is to substitute the coordinates into the circle's equation and see if the result satisfies the inequality:

[(4 - 1)^2 (2 - 4)^2 13]

Since (13

Conclusion

Based on the calculations, we conclude that the point ((4, 2)) lies inside the circle with the equation (x - 1^2 y - 4^2 22). This conclusion is verified by both the distance calculation from the center and the substitution into the circle's equation.

Additional Insights

1. **Standard Circle Equation**: The equation (x - 1^2 y - 4^2 22) represents a circle with center ((1, 4)) and radius (sqrt{22}).

2. **Distance Calculation**: Using the distance formula, we find the distance from the point ((4, 2)) to the center ((1, 4)) is (sqrt{13}).

3. **Inequality Check**: Substituting ((4, 2)) into the circle's equation results in (13

4. **Point on the Circle**: A point would lie on the circle if the distance from the center equaled the radius, i.e., (d sqrt{22}).

By understanding and applying these methods, you can easily determine the position of any point relative to a given circle.