For what value of [tex]$\lambda$[/tex] will the point [tex]$(2,3)$[/tex] lie on the locus whose equation is [tex]$x^2 + y^2 + \lambda x + 2y - 30 = 0$[/tex]?



Answer :

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].