To solve the given problem regarding the circle [tex]\( C \)[/tex] with the equation [tex]\( x^2 + y^2 - 6x + 10y + k = 0 \)[/tex]:
### Part (a)
To find the coordinates of the center of the circle, we need to rewrite the given equation in the standard form of a circle’s equation, [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex]. We can achieve this by completing the square for the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] terms.
Let's start by grouping the [tex]\( x \)[/tex] and [tex]\( y \)[/tex] terms:
[tex]\[ x^2 - 6x + y^2 + 10y + k = 0 \][/tex]
#### Completing the square for [tex]\( x \)[/tex]:
[tex]\[ x^2 - 6x \][/tex]
To complete the square, we add and subtract [tex]\((\frac{-6}{2})^2 = 9\)[/tex]:
[tex]\[ x^2 - 6x = (x - 3)^2 - 9 \][/tex]
#### Completing the square for [tex]\( y \)[/tex]:
[tex]\[ y^2 + 10y \][/tex]
To complete the square, we add and subtract [tex]\((\frac{10}{2})^2 = 25\)[/tex]:
[tex]\[ y^2 + 10y = (y + 5)^2 - 25 \][/tex]
Substituting back into the original equation:
[tex]\[ (x - 3)^2 - 9 + (y + 5)^2 - 25 + k = 0 \][/tex]
Simplify:
[tex]\[ (x - 3)^2 + (y + 5)^2 - 34 + k = 0 \][/tex]
Rearrange to express it in standard form:
[tex]\[ (x - 3)^2 + (y + 5)^2 = 34 - k \][/tex]
From this, we can see that the center of the circle is [tex]\((h, k)\)[/tex]:
[tex]\[ (h, k) = (3, -5) \][/tex]
So, the coordinates of the center are [tex]\((3, -5)\)[/tex].
### Part (b)
Given that the circle does not cut or touch the [tex]\( x \)[/tex]-axis, it means its radius must be less than the distance from the center to the [tex]\( x \)[/tex]-axis.
The [tex]\( y \)[/tex]-coordinate of the center is [tex]\(-5\)[/tex], so its distance from the [tex]\( x \)[/tex]-axis is 5 units.
The radius squared [tex]\( r^2 \)[/tex] from the standard form of the circle's equation is [tex]\( 34 - k \)[/tex].
For the circle to not touch the [tex]\( x \)[/tex]-axis, the radius must be less than 5. Thus:
[tex]\[ 34 - k < 25 \][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[ 34 - k < 25 \][/tex]
[tex]\[ -k < 25 - 34 \][/tex]
[tex]\[ -k < -9 \][/tex]
[tex]\[ k > 9 \][/tex]
Therefore, the range of possible values for [tex]\( k \)[/tex] is:
[tex]\[ k > 9 \][/tex]
So, the final solution is:
- The coordinates of the center of circle [tex]\( C \)[/tex] are [tex]\((3, -5)\)[/tex].
- The range of possible values for [tex]\( k \)[/tex] is [tex]\( k > 9 \)[/tex].
