Write the given equation in the form $(x-h)^2+(y-k)^2=r^2$. Identify the center and radius.
7) $x^2+y^2+12x-6y-55=0$
A) $(x+6)^2+(y-3)^2=100$
Center: $(-6,3) ; r=10$
B) $(x+6)^2+$
C) $(x+12)^2+(y-6)^2=55$
Center: $(6,$
Center: $(12,-6) ; r=55$
D) $(x+12)^2$
Center: (-
Find the domain in interval notation.
8) $f(x)=\frac{8}{\sqrt{7-x}}$
A) $(-\infty, 7]$
B) $[7, \infty)$
C) $(7, \infty)$
Answer (2)
Convert the circle equation to standard form by completing the square: ( x + 6 ) 2 + ( y − 3 ) 2 = 100 .
Identify the center and radius: Center ( − 6 , 3 ) , radius r = 10 .
Determine the domain of f ( x ) = 7 − x 8 by solving 0"> 7 − x > 0 .
Express the domain in interval notation: ( − ∞ , 7 ) . The final answers are A and D respectively.
Explanation
Convert to Standard Form We are given the equation of a circle in general form: x 2 + y 2 + 12 x − 6 y − 55 = 0 . We need to convert it to the standard form ( x − h ) 2 + ( y − k ) 2 = r 2 , where ( h , k ) is the center and r is the radius. To do this, we complete the square for both x and y terms.
Group terms and prepare to complete the square First, group the x and y terms together: ( x 2 + 12 x ) + ( y 2 − 6 y ) = 55 . To complete the square for x 2 + 12 x , we need to add and subtract ( 2 12 ) 2 = 6 2 = 36 . To complete the square for y 2 − 6 y , we need to add and subtract ( 2 − 6 ) 2 = ( − 3 ) 2 = 9 .
Add and subtract values to complete the square Now, add and subtract these values within the equation: ( x 2 + 12 x + 36 − 36 ) + ( y 2 − 6 y + 9 − 9 ) = 55 . Rewrite as ( x 2 + 12 x + 36 ) + ( y 2 − 6 y + 9 ) = 55 + 36 + 9 .
Factor and simplify Factor the perfect square trinomials: ( x + 6 ) 2 + ( y − 3 ) 2 = 100 . This is now in the standard form ( x − h ) 2 + ( y − k ) 2 = r 2 .
Identify center and radius From the standard form, we can identify the center as ( h , k ) = ( − 6 , 3 ) and the radius as r = 100 = 10 . Therefore, the correct answer is A) ( x + 6 ) 2 + ( y − 3 ) 2 = 100 with center ( − 6 , 3 ) and r = 10 .
Find the domain Now, let's find the domain of the function f ( x ) = 7 − x 8 . The domain is the set of all possible x values for which the function is defined. Since we have a square root in the denominator, the expression inside the square root must be strictly greater than 0.
Solve the inequality and express in interval notation So, we have the inequality 0"> 7 − x > 0 . Adding x to both sides gives x"> 7 > x , or x < 7 . In interval notation, this is ( − ∞ , 7 ) . Therefore, the correct answer is D) ( − ∞ , 7 ) .
Examples
Understanding circles and their equations is crucial in many fields, such as physics and engineering. For example, when designing a circular garden or a satellite dish, knowing the center and radius helps in accurate construction and placement. Similarly, determining the domain of a function is essential in modeling real-world phenomena, ensuring that the model produces meaningful results within a specific range of inputs. For instance, when modeling population growth, the domain might represent the time period for which the model is valid, ensuring that the population remains non-negative and realistic.
The equation of the circle can be rewritten in standard form as ( x + 6 ) 2 + ( y − 3 ) 2 = 100 , which identifies the center as ( − 6 , 3 ) and the radius as 10 . The domain of the function f ( x ) = 7 − x 8 is ( − ∞ , 7 ) . Thus, the answers are A for the circle equation and D for the domain.
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