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Blababa [14]
2 years ago
10

29. (09.04 MC)

Mathematics
1 answer:
lara [203]2 years ago
7 0

Answer:

The center is located at (-1, -2), and the radius is 5.

Step-by-step explanation:

The general equation of a circle is given as

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

where

a and b are the centre of the circle and r is the radius

Given the equation

x2 + 2x + y2 + 4y = 20

add 1 ad 4 to both sides

x2 + 2x + 1 + y2 + 4y + 4 = 20 + 4 + 1

factorise

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

comparing with the general equation

a = -1, b = -2, r = 5 (all measured in units)

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Given <br><img src="https://tex.z-dn.net/?f=%20log_%7B2%7D%28x%29%20%20%3D%20%20%5Cfrac%7B3%7D%7B%20log_%7Bxy%7D%282%29%20%7D%20
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Answer:

\displaystyle y = x^{-\frac{2}{3}}

Step-by-step explanation:

<u>Logarithms</u>

Some properties of logarithms will be useful to solve this problem:

1. \log(pq)=\log p+\log q

2. \displaystyle \log_pq=\frac{1}{\log_qp}

3. \displaystyle \log p^q=q\log p

We are given the equation:

\displaystyle \log_{2}(x) = \frac{3}{ \log_{xy}(2) }

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\displaystyle  \log_{xy}(2)=\frac{1}{ \log_{2}(xy)}

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\displaystyle \log_{2}(x) = 3\log_{2}(xy)

Applying the first property:

\displaystyle \log_{2}(x) = 3(\log_{2}(x)+\log_{2}(y))

Operating:

\displaystyle \log_{2}(x) = 3\log_{2}(x)+3\log_{2}(y)

Rearranging:

\displaystyle \log_{2}(x) - 3\log_{2}(x)=3\log_{2}(y)

Simplifying:

\displaystyle -2\log_{2}(x) =3\log_{2}(y)

Dividing by 3:

\displaystyle \log_{2}(y)=\frac{-2\log_{2}(x)}{3}

Applying the third property:

\displaystyle \log_{2}(y)=\log_{2}\left(x^{-\frac{2}{3}}\right)

Applying inverse logs:

\boxed{y = x^{-\frac{2}{3}}}

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