To determine the value of [tex]\(\lambda\)[/tex] for which the point [tex]\((2, 3)\)[/tex] lies on the locus defined by the equation [tex]\(x^2 + y^2 + \lambda x + 2y - 30 = 0\)[/tex], proceed with the following steps:
1. Identify the coordinates of the point:
The given point is [tex]\((x, y) = (2, 3)\)[/tex].
2. Substitute the coordinates into the equation:
Plug in [tex]\(x = 2\)[/tex] and [tex]\(y = 3\)[/tex] into the equation:
[tex]\[
(2)^2 + (3)^2 + \lambda(2) + 2(3) - 30 = 0
\][/tex]
3. Simplify the equation:
Calculate the individual terms:
[tex]\[
4 + 9 + 2\lambda + 6 - 30 = 0
\][/tex]
Combine the constants:
[tex]\[
19 + 2\lambda - 30 = 0
\][/tex]
Simplify further:
[tex]\[
2\lambda - 11 = 0
\][/tex]
4. Solve for [tex]\(\lambda\)[/tex]:
Isolate the variable [tex]\(\lambda\)[/tex]:
[tex]\[
2\lambda = 11
\][/tex]
Divide both sides by 2:
[tex]\[
\lambda = \frac{11}{2} = 5.5
\][/tex]
Therefore, the value of [tex]\(\lambda\)[/tex] for which the point [tex]\((2, 3)\)[/tex] lies on the given locus is [tex]\(\lambda = 5.5\)[/tex].
