Question

Ki are possible values of K for which lines $Kx + 2y + 2 = 0$, $2x + Ky + 3 = 0$, $3x + 3y + K = 0$ are concurrent, then $∑k_i$ has value.

• A. 0
• B. -2
• C. 2
• D. 5

Answer 1

Answer: (A)

0

Answer 2

To determine the possible values of K for which the lines $Kx + 2y + 2 = 0$, $2x + Ky + 3 = 0$, and $3x + 3y + K = 0$ are concurrent, we need to check if the determinant of the coefficient matrix is zero.

The coefficient matrix is: $\begin{vmatrix} K & 2 & 0\\\ 2 & K & 3\\\ 3 & 3 & K \end{vmatrix}$

Taking the determinant of this matrix and setting it equal to zero, we get:

$K(K^2 - 6) - 6(K - 6) + 18(3 - 3K) = 0$

Simplifying the equation, we have: $K^3 - 6K^2 - 6K + 216 = 0$

Factoring the left-hand side, we find: $(K - 6)(K^2 + 12K - 36) = 0$

Setting each factor equal to zero, we obtain two possible values for K: $K = 6$ and $K = -6 ± 2\sqrt3$

The sum of these possible values of K is: $6 + (-6 + 2\sqrt3) + (-6 - 2\sqrt3) = 6 - 6 = 0$

Therefore, the value of $∑k_i$ is 0.