What is the standard equation of a circle?

Question

I have no idea please help

Answer 1

Lovely,

 

If your circle of radius r is centered at point (x,y)=(a,b), then the standard equation of the circle in coordinate system x,y is:

 

(x-a)²  +  (y-b)²  =  r² . 

 

For example,

(x+7)²  +  (y-4)²  = 25  = (5)² .  represents a circle centered at point (-7,+4) with radius 5.

 

NOTE.  If the circle is centered at the origin (then a=b=0), and you have the familiar formula for a right triangle: hypotenuse squared = side-x squared plus side-y squared.  The hypotenuse is the line segment from (0,0) to (x,y) on the circle's circumference. Side-x is segment between (0,0) and (x,0), and side-y is the segment between (0,0) and (0,y).  Then as the hypotenuse is rotated across all angles, the triangle remains a right triangle but whose x & y sides change lengths to maintain a constant hypotenuse length (r).

 

For a circle, the two terms on the left side of the standard form must be squared and the sign between them must always  be positive.  If the sign was negative, then the radius would not be constant so it could not be a circle.  Some people divide the above equation through by r², and thereby divide each term by r², leaving the right side as 1.  Then it will become:

 

[(x-a)/r]²  +  [(y-b)/r]²  =  1

 

Then the equation of an ellipse will look similar except it will have different r-values (r_a & r_b) for the x & y terms.

 

 

Answer 2

(x – h)² + (y – k)² = r², with the center being at the point (h, k) and the radius being r.

 

Ex. (x – 2)² + (y + 3)² = 16

Center (2, -3), radius 4