central equation: x² + y² = r²

First, if we keep the radius r constant, then, as the magnitude of x increases, the magnitude of y must decrease. We express this like so:

(x)² ↑ + (y)² ↓= r².

Arrows appended to variables indicate increase and decrease; all other values are understood to be constant. (Sometimes a horizontal arrow is placed above constant terms to indicate they are constant. Here we will understand that these terms without arrows are held constant.)

Note:  given    4  +  16  = 20,   if 4 increases to 12, then 16 must decrease to 8 to maintain 20 constant.  12 + 8 = 20

Likewise if the magnitude of x decreases, then the magnitude of y must increase.

(x)² ↓ + (y)² ↑= r².

In the graph below you can see these effects. Original values are in black, with the new values in red. The subscripts 0 and 1 for x and y represent original and new values respectively.

Note the length of the radius is constant, and is shown as the blue dashed lines from the centers of the circles to the perimeters of the circles. The radius must 'move' that is, rotate, to maintain the relationship between x and y. This movement is shown with the heavy blue arrows.

Using the first graph we start with the pair (x₀, y₀). Now increase x₀ to x₁. The new pair (x₁, y₁) must lie on the circle (otherwise it does not satisfy the equation for this circle.)

For this to be true, y must decrease from y₀ to y₁. Likewise, if y decreases from y₀ to y₁ then x must increase from x₀ to x₁.

The second graph shows the opposite relationships. If x decreases from x₀ to x₁ then y must increase from y₀ to y_{1 .} And, if y₀ increases to y₁ then x₀ decreases to x₁.

(Understanding this relationship between x and y is crucial to understanding the unit circle and the trigonometric functions in Trigonometry. Study this well, now.)

If x is held constant and r increases then the magnitude of y must also increase and if r decreases than magnitude of y must decrease. Likewise if y is held constant then if r decreases then the magnitude x must decrease.

The radius must 'rotate' into a new position, shown by the red arrows. In the first case it moves until its endpoint intersects the vertical line x_0. In the second case it moves until it intersects the horizontal line y₀.

First case:

(x)² + (y)² ↑= (r²)↑

Note:  given    4  +  16  = 20,   if 16 increases to 30, then 20 must increase to 34 to maintain 4 constant.  4 + 30 = 34

Look at the graph on the left this way: The black radius is the starting point, it intersects the black circle at (x₀,y₀). Now extend this radius by the amount shown by the red dashed line. This radius now intersects the red circle but its x coordinate has changed, it has become larger than the original x₀. We want to keep the x value constant, and the only way we can do this is to rotate the black radius counterclockwise to the position of the red radius. This movement maintains the value x₀ but increases the y value by the amount shown with the vertical dark arrow from y₀ to y₁.

Second case:

(x)² ↓ + (y)² = ( r²) ↓

Explanation: (try this one on your own then compare to this explanation.)

Starting with the black circle, the black radius intersects this circle at (x₀, y₀).

In this case we are holding y constant and decreasing the radius, r.

When we decrease the black radius by the amount shown with the red dashed line we must rotate the black radius counterclockwise to maintain the constant y value. This movement creates the smaller red circle and the intersection of this new red radius requires a smaller x value. This means that x must decrease to satisfy the equation for the circle with this constant y value and smaller radius.