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Sagot :
Sure, let's solve the given problem step by step. We are given the equation of a circle:
[tex]\[ x^2 + y^2 + 8x - 4y + k = 0 \][/tex]
and we need to find the coordinates of the center of the circle and the value of [tex]\( k \)[/tex].
### Part (a): Finding the coordinates of the center of the circle
To find the center, we first need to rewrite the given equation in the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex]. We do this by completing the square for both [tex]\( x \)[/tex] and [tex]\( y \)[/tex] terms.
1. Complete the square for the [tex]\( x \)[/tex]-terms:
The [tex]\( x \)[/tex]-terms in the equation are [tex]\( x^2 + 8x \)[/tex].
- Take the coefficient of [tex]\( x \)[/tex], which is 8, divide it by 2 to get 4, and then square it to get 16.
- Add and subtract 16 to complete the square.
So, [tex]\( x^2 + 8x \rightarrow (x + 4)^2 - 16 \)[/tex].
2. Complete the square for the [tex]\( y \)[/tex]-terms:
The [tex]\( y \)[/tex]-terms in the equation are [tex]\( y^2 - 4y \)[/tex].
- Take the coefficient of [tex]\( y \)[/tex], which is -4, divide it by 2 to get -2, and then square it to get 4.
- Add and subtract 4 to complete the square.
So, [tex]\( y^2 - 4y \rightarrow (y - 2)^2 - 4 \)[/tex].
3. Rewrite the equation with completed squares:
Combining the completed squares, we get:
[tex]\[ (x + 4)^2 - 16 + (y - 2)^2 - 4 + k = 0 \][/tex]
[tex]\[ (x + 4)^2 + (y - 2)^2 - 20 + k = 0 \][/tex]
[tex]\[ (x + 4)^2 + (y - 2)^2 = 20 - k \][/tex]
Thus, the standard form of the circle's equation is:
[tex]\[ (x + 4)^2 + (y - 2)^2 = r^2 \][/tex]
From this equation, we can see that the center of the circle is at:
[tex]\[ (h, k) = (-4, 2) \][/tex]
### Part (b): Finding the value of [tex]\( k \)[/tex]
Given that the [tex]\( x \)[/tex]-axis is a tangent to the circle, the distance from the center of the circle to the x-axis must be equal to the radius of the circle [tex]\( r \)[/tex].
1. Find the radius:
The radius [tex]\( r \)[/tex] is the distance from the center of the circle to the [tex]\( x \)[/tex]-axis. Since the [tex]\( x \)[/tex]-axis is a tangent and the center's [tex]\( y \)[/tex]-coordinate is 2, the distance (or radius) [tex]\( r \)[/tex] must be 2.
2. Determine the value of [tex]\( k \)[/tex]:
We know the radius [tex]\( r = 2 \)[/tex], so [tex]\( r^2 = 4 \)[/tex].
Using the standard form equation:
[tex]\[ (x + 4)^2 + (y - 2)^2 = r^2 \][/tex]
and substituting [tex]\( r^2 = 4 \)[/tex], we get:
[tex]\[ (x + 4)^2 + (y - 2)^2 = 4 \][/tex]
We also have:
[tex]\[ (x + 4)^2 + (y - 2)^2 = 20 - k \][/tex]
Setting the above expressions equal to each other gives:
[tex]\[ 4 = 20 - k \][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[ k = 20 - 4 \][/tex]
[tex]\[ k = 16 \][/tex]
So the value of [tex]\( k \)[/tex] is:
[tex]\[ k = 16 \][/tex]
Therefore, the coordinates of the center of the circle are:
[tex]\[ (-4, 2) \][/tex]
and the value of [tex]\( k \)[/tex] is:
[tex]\[ 16 \][/tex]
[tex]\[ x^2 + y^2 + 8x - 4y + k = 0 \][/tex]
and we need to find the coordinates of the center of the circle and the value of [tex]\( k \)[/tex].
### Part (a): Finding the coordinates of the center of the circle
To find the center, we first need to rewrite the given equation in the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex]. We do this by completing the square for both [tex]\( x \)[/tex] and [tex]\( y \)[/tex] terms.
1. Complete the square for the [tex]\( x \)[/tex]-terms:
The [tex]\( x \)[/tex]-terms in the equation are [tex]\( x^2 + 8x \)[/tex].
- Take the coefficient of [tex]\( x \)[/tex], which is 8, divide it by 2 to get 4, and then square it to get 16.
- Add and subtract 16 to complete the square.
So, [tex]\( x^2 + 8x \rightarrow (x + 4)^2 - 16 \)[/tex].
2. Complete the square for the [tex]\( y \)[/tex]-terms:
The [tex]\( y \)[/tex]-terms in the equation are [tex]\( y^2 - 4y \)[/tex].
- Take the coefficient of [tex]\( y \)[/tex], which is -4, divide it by 2 to get -2, and then square it to get 4.
- Add and subtract 4 to complete the square.
So, [tex]\( y^2 - 4y \rightarrow (y - 2)^2 - 4 \)[/tex].
3. Rewrite the equation with completed squares:
Combining the completed squares, we get:
[tex]\[ (x + 4)^2 - 16 + (y - 2)^2 - 4 + k = 0 \][/tex]
[tex]\[ (x + 4)^2 + (y - 2)^2 - 20 + k = 0 \][/tex]
[tex]\[ (x + 4)^2 + (y - 2)^2 = 20 - k \][/tex]
Thus, the standard form of the circle's equation is:
[tex]\[ (x + 4)^2 + (y - 2)^2 = r^2 \][/tex]
From this equation, we can see that the center of the circle is at:
[tex]\[ (h, k) = (-4, 2) \][/tex]
### Part (b): Finding the value of [tex]\( k \)[/tex]
Given that the [tex]\( x \)[/tex]-axis is a tangent to the circle, the distance from the center of the circle to the x-axis must be equal to the radius of the circle [tex]\( r \)[/tex].
1. Find the radius:
The radius [tex]\( r \)[/tex] is the distance from the center of the circle to the [tex]\( x \)[/tex]-axis. Since the [tex]\( x \)[/tex]-axis is a tangent and the center's [tex]\( y \)[/tex]-coordinate is 2, the distance (or radius) [tex]\( r \)[/tex] must be 2.
2. Determine the value of [tex]\( k \)[/tex]:
We know the radius [tex]\( r = 2 \)[/tex], so [tex]\( r^2 = 4 \)[/tex].
Using the standard form equation:
[tex]\[ (x + 4)^2 + (y - 2)^2 = r^2 \][/tex]
and substituting [tex]\( r^2 = 4 \)[/tex], we get:
[tex]\[ (x + 4)^2 + (y - 2)^2 = 4 \][/tex]
We also have:
[tex]\[ (x + 4)^2 + (y - 2)^2 = 20 - k \][/tex]
Setting the above expressions equal to each other gives:
[tex]\[ 4 = 20 - k \][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[ k = 20 - 4 \][/tex]
[tex]\[ k = 16 \][/tex]
So the value of [tex]\( k \)[/tex] is:
[tex]\[ k = 16 \][/tex]
Therefore, the coordinates of the center of the circle are:
[tex]\[ (-4, 2) \][/tex]
and the value of [tex]\( k \)[/tex] is:
[tex]\[ 16 \][/tex]
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