Transformation of Circle A: Equation After Shifts and Radius Doubling

The problem at hand deals with transforming the given circle equation through a series of operations, including shifting and doubling the radius. This article will walk through the steps to modify the initial circle equation and derive the new equation after the transformations.

Original Circle Equation

The original equation of Circle A is as follows:

x - 4^2 y 3^2 29

From this equation, we can identify the center and the radius of the circle. The standard form of a circle's equation is:

(x - h)^2 (y - k)^2 r^2

Comparing the given equation to this format, we can see:

(x - 4)^2 (y - 3)^2 29

This indicates that the center of Circle A is at (4, -3) and the radius is sqrt{29}.

Shifting the Circle

The circle is then shifted according to the given operations:

Five units up in the y-direction. Six units left in the x-direction.

To find the new center, we simply adjust the coordinates for each shift:

Shifting up 5 units changes the y-coordinate from -3 to -3 5 2. Shifting left 6 units changes the x-coordinate from 4 to 4 - 6 -2.

Thus, the new center after the shifts is at (-2, 2).

Doubling the Radius

The radius of the circle is then doubled, which means:

The new radius is 2 * sqrt{29}.

Remember, if the original radius is r, then the new radius is 2r.

Forming the New Equation

The general form of the circle's equation with the new center and radius is:

(x - h)^2 (y - k)^2 (2r)^2

Substituting the new coordinates for the center and the value for the new radius:

h -2 k 2 2r 2 * sqrt{29} (2r)^2 (2 * sqrt{29})^2 4 * 29 116

The equation of the new circle is:

(x - (-2))^2 (y - 2)^2 116

Simplifying:

(x 2)^2 (y - 2)^2 116

Conclusion

In summary, the original circle with the equation x - 4^2 y 3^2 29, after shifting up by 5 units and left by 6 units, and then doubling its radius, transforms to the new equation:

(x 2)^2 (y - 2)^2 116

Understanding the transformations of circle equations is crucial, especially in fields such as geometry and computer graphics. This process not only helps in visualizing but also in applying advanced mathematical concepts.