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

  • Thread starter jeahomgrajan
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In summary, the implicit equation x^2 + y^2 = 25 maps to a circle with a radius of 5 and the equation y = sqrt(25 - x^2) produces the upper half of the circle while y = -sqrt(25 - x^2) produces the lower half. The equation y = 5 - x does not make a semi-circle, but rather a line.
  • #1
jeahomgrajan
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Homework Statement



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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  • #2
If you would solve [tex]x^{2} + y^{2} = 25[/tex] for y, you would not only get [tex]y=\sqrt{25-x^{2}}[/tex], but also something else. What?
 
  • #3
y=5-x
 
  • #4
jeahomgrajan said:
y=5-x
No, not quite. (Keep in mind that [tex]\sqrt{a^{2}-b^{2}}\neq a-b[/tex].)

Think of something as simple as e.g. [tex]x^{2}=9[/tex]. There are two different values of x that satisfy this. What values would that be?
 
  • #5
+-3?
 
  • #6
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.
 
  • #7
intence, but okay, what about the second equation which i have mentioned
 
  • #8
jeahomgrajan said:
y=5-x ( that would make a semi circle right)

y = 5-x is a line, how are you getting semicircle
 
  • #9
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 [tex]y^{2}=25-x^{2}[/tex], 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 = [itex]-\sqrt{25 - x^2}[/itex]
 
  • #13
What people were trying to tell you was that if [tex] a = b^2 [/tex] then [tex] b = \pm \sqrt{a} [/tex] NOT just [tex] \sqrt{a} [/tex]
 
  • #14
Alright i understand thanks
 

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

1. What is the equation for the graph x^2 + y^2 = 25?

The equation x^2 + y^2 = 25 represents a circle with a radius of 5 and a center at the origin (0,0).

2. What are the solutions to the equation y = sqrt(25-x^2)?

The solutions to this equation are all real values of x that make the expression under the square root positive. This includes all values of x between -5 and 5, as well as the two endpoints (x = -5 and x = 5).

3. How does changing the value of x affect the graph of x^2 + y^2 = 25?

Changing the value of x will shift the graph of this equation horizontally. As x increases, the circle will shift to the right, and as x decreases, the circle will shift to the left.

4. What is the relationship between the equations x^2 + y^2 = 25 and y = sqrt(25-x^2)?

The first equation represents a circle with a radius of 5 and a center at the origin. The second equation represents a semicircle with the same radius and center, but only the top half is graphed due to the square root function. Both equations have the same solutions, but they have different graphs.

5. Can you solve the system of equations x^2 + y^2 = 25 and y = sqrt(25-x^2)?

Yes, we can solve this system of equations graphically by finding the points where the circle and semicircle intersect. The solutions are (0,5) and (0,-5).

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