Let's solve the given problem step-by-step:
### (a) Find the value of [tex]\(k\)[/tex]:
We are given the circle's equation:
[tex]\[ x^2 + y^2 + 10x - 8y + k = 0 \][/tex]
And we know that the point [tex]\((-6, 5)\)[/tex] lies on this circle. Substitute [tex]\(x = -6\)[/tex] and [tex]\(y = 5\)[/tex] into the equation:
[tex]\[ (-6)^2 + (5)^2 + 10(-6) - 8(5) + k = 0 \][/tex]
Calculate each term:
[tex]\[ 36 + 25 - 60 - 40 + k = 0 \][/tex]
Simplify the equation:
[tex]\[ 36 + 25 = 61 \][/tex]
[tex]\[ 61 - 60 = 1 \][/tex]
[tex]\[ 1 - 40 = -39 \][/tex]
[tex]\[ -39 + k = 0 \][/tex]
Solve for [tex]\(k\)[/tex]:
[tex]\[ k = 39 \][/tex]
### (b) Find the coordinates of the center and the radius of the circle:
Rewrite the circle equation in the standard form by completing the square. Starting with:
[tex]\[ x^2 + y^2 + 10x - 8y + 39 = 0 \][/tex]
Group the [tex]\(x\)[/tex]-terms and [tex]\(y\)[/tex]-terms:
[tex]\[ (x^2 + 10x) + (y^2 - 8y) = -39 \][/tex]
Complete the square for each group:
[tex]\[ x^2 + 10x \rightarrow (x + 5)^2 - 25 \][/tex]
[tex]\[ y^2 - 8y \rightarrow (y - 4)^2 - 16 \][/tex]
So, rewrite the equation:
[tex]\[ (x + 5)^2 - 25 + (y - 4)^2 - 16 = -39 \][/tex]
Combine constants on the right-hand side:
[tex]\[ (x + 5)^2 + (y - 4)^2 - 41 = -39 \][/tex]
[tex]\[ (x + 5)^2 + (y - 4)^2 = 2 \][/tex]
From this equation, it's clear that the circle's center is [tex]\((-5, 4)\)[/tex] and the radius squared is 2, so the radius is:
[tex]\[ \text{Radius} = \sqrt{2} \][/tex]
### (c) Find the length [tex]\(AB\)[/tex]:
We are given that a line passes through the point [tex]\(A(2, 3)\)[/tex] and is tangent to the circle at point [tex]\(B\)[/tex]. The distance from point [tex]\(A\)[/tex] to the point of tangency [tex]\(B\)[/tex] can be found using the length of the tangent from a point to a circle.
Given:
- The circle's radius is [tex]\(\sqrt{2}\)[/tex].
- The center of the circle is [tex]\((-5, 4)\)[/tex].
The length of the tangent from a point [tex]\((x_1, y_1)\)[/tex] to a circle with radius [tex]\(r\)[/tex] and center [tex]\((h, k)\)[/tex] is:
[tex]\[ \text{Length of tangent} = \sqrt{(x_1 - h)^2 + (y_1 - k)^2 - r^2} \][/tex]
Substitute the values [tex]\(A(2, 3)\)[/tex] and [tex]\((-5, 4)\)[/tex]:
[tex]\[ \text{Length} = \sqrt{(2 + 5)^2 + (3 - 4)^2 - (\sqrt{2})^2} \][/tex]
[tex]\[ = \sqrt{7^2 + (-1)^2 - 2} \][/tex]
[tex]\[ = \sqrt{49 + 1 - 2} \][/tex]
[tex]\[ = \sqrt{48} \][/tex]
[tex]\[ = 4\sqrt{3} \][/tex]
So, the length [tex]\(AB\)[/tex] is:
[tex]\[ AB = 4\sqrt{3} \][/tex]
Summarizing the answers:
1. The value of [tex]\(k\)[/tex] is [tex]\(39\)[/tex].
2. The center of the circle is [tex]\((-5, 4)\)[/tex], and the radius is [tex]\(\sqrt{2}\)[/tex].
3. The length [tex]\(AB\)[/tex] is [tex]\(4\sqrt{3}\)[/tex].
