Uncovering the Geometry of x^2+y^2=25 & y=sqrt 25-x^2

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The equation x^2 + y^2 = 25 represents a circle with a radius of 5, as it defines all points equidistant from the center. When solving for y, both y = √(25 - x^2) and y = -√(25 - x^2) are obtained, indicating the upper and lower halves of the circle, respectively. The equation y = 5 - x is incorrectly interpreted as a semicircle; it is actually a linear equation. The correct interpretation of the semicircle comes from y = √(25 - x^2), which represents the upper half of the circle. Understanding the distinction between positive and negative square roots is crucial in this context.
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This is a simple math equation and i am a bit confused on how

x^2+y^2=25 makes a circle and how y= sqrt 25-x^2 makes a semi circle
 
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If you would solve x^{2} + y^{2} = 25 for y, you would not only get y=\sqrt{25-x^{2}}, but also something else. What?
 
y=5-x
 
jeahomgrajan said:
y=5-x
No, not quite. (Keep in mind that \sqrt{a^{2}-b^{2}}\neq a-b.)

Think of something as simple as e.g. x^{2}=9. There are two different values of x that satisfy this. What values would that be?
 
+-3?
 
Well, since the circle is defined as the locus of all points in a plane which are all a fixed and equal distance from the center, it is fairly intuitively obvious that the implicit equation maps to a circle. A more general case of the formula you presented is x^2 + y^2 = r^2 where r is the radius of the circle. Perhaps a general way to derive and illustrate how that equation makes a circle is to examine the circular definition of the trigonometric functions. We know that sin(theta) = y/r and cos(theta) = x/r where r is the radius and (x,y) is the coordinate of the radius' intersection with the circle. I'm also presuming you're familiar with the identity sin^2(theta) + cos^2(theta) = 1. Placing our unit circle definitions in place of the identity yields x^2 + y^2 = r^2 which may intuitively illustrate how it makes a circle.

Probably a better way of thinking about it is this: given any ordered pair (x,y) satisfying x^2 + y^2 = r^2, (x,y) must always be some fixed distance r from the center; hence, the equation produces the locus of all points on a plane an equal distance from the center. In other words, x^2 + y^2 = r^2 maps to a circle.
 
intence, but okay, what about the second equation which i have mentioned
 
jeahomgrajan said:
y=5-x ( that would make a semi circle right)

y = 5-x is a line, how are you getting semicircle
 
y = 5 - x will definitely not make a semi-circle. Solve x^2 + y^2 = r^2 for y and it should be fairly clear why it produces a semi-circle if x^2 + y^2 = r^2 creates a circle.
 
  • #10
jeahomgrajan said:
+-3?

Exactly. So if you now have y^{2}=25-x^{2}, what two values of y do you get?
 
  • #11
y= SqaUREROOT 25-x^2, (this should be a semicircle right?
 
  • #12
Yes, this is the upper half of the circle. The lower half is y = -\sqrt{25 - x^2}
 
  • #13
What people were trying to tell you was that if a = b^2 then b = \pm \sqrt{a} NOT just \sqrt{a}
 
  • #14
Alright i understand thanks
 
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