To determine the value of [tex]\( k \)[/tex] such that the point [tex]\((2, 1)\)[/tex] lies on the locus described by the equation [tex]\( x^2 + y^2 - 4x + 3y + k = 13 \)[/tex], follow these steps:
1. Substitute the coordinates [tex]\( (2, 1) \)[/tex] into the equation:
Given [tex]\( x = 2 \)[/tex] and [tex]\( y = 1 \)[/tex], substitute these values into the equation:
[tex]\[
x^2 + y^2 - 4x + 3y + k = 13
\][/tex]
will become:
[tex]\[
(2)^2 + (1)^2 - 4(2) + 3(1) + k = 13
\][/tex]
2. Simplify the equation:
- Calculate [tex]\( (2)^2 \)[/tex]:
[tex]\[
(2)^2 = 4
\][/tex]
- Calculate [tex]\( (1)^2 \)[/tex]:
[tex]\[
(1)^2 = 1
\][/tex]
- Calculate [tex]\(-4(2)\)[/tex]:
[tex]\[
-4(2) = -8
\][/tex]
- Calculate [tex]\( 3(1) \)[/tex]:
[tex]\[
3(1) = 3
\][/tex]
Substitute these values back into the equation:
[tex]\[
4 + 1 - 8 + 3 + k = 13
\][/tex]
3. Combine like terms:
First, combine the numeric values:
[tex]\[
4 + 1 = 5
\][/tex]
Then:
[tex]\[
5 - 8 = -3
\][/tex]
Finally:
[tex]\[
-3 + 3 = 0
\][/tex]
This leaves us with:
[tex]\[
0 + k = 13
\][/tex]
4. Isolate [tex]\( k \)[/tex]:
[tex]\[
k = 13
\][/tex]
Therefore, the value of [tex]\( k \)[/tex] that ensures the point [tex]\((2, 1)\)[/tex] lies on the locus given by the equation [tex]\( x^2 + y^2 - 4x + 3y + k = 13 \)[/tex] is [tex]\( k = 13 \)[/tex].
