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quester [9]
3 years ago
14

The equation x^2+y^2-4x+2y=b describes a circle. Determine the y-coordinate of the center of the circle.

Mathematics
2 answers:
abruzzese [7]3 years ago
8 0

Answer:

Step-by-step explanation:

The equation is given as:

x^2+y^2-4x+2y=b

Upon solving this equation, we have

x^2-4x+4+y^2+2y+1=b+4+1

which can be written as:

(x-2)^2+(y+1)^2=b+5                   (1)

Thus, the y- coordinate of the center of the circle is y=-1.

Now, comparing the equation (1) with the equation of circle, we have

(x-a)^2+(y-c)^2=r^2

where r is the radius of the circle and (a,c) is the center.

Thus, on comparing, we have

⇒b+5=r^2

Also, it is given that the radius of the circle is 7 units, thus putting r=7 in above equation, we get

⇒b+5=(7)^2

⇒b+5=49

⇒b=44 units

Thus, the value of b is 44 units in the given equation.

irga5000 [103]3 years ago
3 0
X² + y² - 4 x + 2 y = b
x² - 4 x + 4 + y² + 2 y + 1 = b + 4 + 1
( x - 2 )² + ( y + 1 )² = b + 5
y - coordinate of the center of the circle is:  y = - 1
b + 5 = r
b + 5 = 7
b = 7 - 5
b = 2
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General Formulas and Concepts:

<u>Calculus</u>

Limits

  • Right-Side Limit:                                                                                             \displaystyle  \lim_{x \to c^+} f(x)
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Step-by-step explanation:

*Note:

In order for a limit to exist, the right-side and left-side limits must equal each other.

<u>Step 1: Define</u>

<em>Identify</em>

\displaystyle f(x) = \left\{\begin{array}{ccc}5 - x,\ x < 5\\8,\ x = 5\\x + 3,\ x > 5\end{array}

<u>Step 2: Find Right-Side Limit</u>

  1. Substitute in function [Limit]:                                                                         \displaystyle  \lim_{x \to 5^+} 5 - x
  2. Evaluate limit [Limit Rule - Variable Direct Substitution]:                           \displaystyle  \lim_{x \to 5^+} 5 - x = 5 - 5 = 0

<u>Step 3: Find Left-Side Limit</u>

  1. Substitute in function [Limit]:                                                                         \displaystyle  \lim_{x \to 5^-} x + 3
  2. Evaluate limit [Limit Rule - Variable Direct Substitution]:                           \displaystyle  \lim_{x \to 5^+} x + 3 = 5 + 3 = 8

∴ Since  \displaystyle \lim_{x \to 5^+} f(x) \neq \lim_{x \to 5^-} f(x)  , then  \displaystyle \lim_{x \to 5} f(x) = DNE

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit:  Limits

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