Question

The equation of a circle is

x2 + y2 + Cx + Dy + E = 0. If the radius of the circle is decreased without changing the coordinates of the center point, how are the coefficients C, D, and E affected?

Answer 1

Decreasing the radius of a circle without changing its center affects the constant E in the circle's equation, while coefficients C and D remain unchanged as they are related to the center's coordinates.

Step-by-step explanation:

The equation of a circle x2 + y2 + Cx + Dy + E = 0 represents a circle in a coordinate plane with a specific center and radius. The coefficients C and D in the equation relate to the coordinates of the center of the circle, whereas the coefficient E is related to the size of the radius of the circle. When you decrease the radius of the circle without changing its center, the value of E changes, but the coefficients C and D remain the same because the center of the circle has not moved.

Understanding that the standard equation of a circle with center at (h, k) and radius r is (x - h)2 + (y - k)2 = r2, it becomes clear that changing the radius will affect the constant term equivalent to r2 but not the terms that define the center which are related to h and k.

Answer 2

first we will write different type of equation for circle so we can see how did we get coeficients C D and E

(x-x1)^2 + (y-y1)^2 = r^2

where x,y is coordinate of a point on circle, x1,y1 are coordinates of the center of circle and r is radius of circle

after solving binomials on square we get terms like this:
-2*x*x1 and -2*y*y1

looking at like term in our given equation we can conclude that there is no "r" in those terms meaning that changing radius doesnt depend on coefficients C and D. They depend only on coordinates of the center of circle.

As for coefficient E.
if we move r^2 to left side we can see that E = x1^2 + y1^2 -r^2 meaning that if we decrease "r", E will increase.