Let c and d be vectors such that | c + d |=√ 29 and c ×(2 i +3 j +4 k )=(2 i +3 j +4 k ) × d. If λ_1, λ_2(λ_1>λ_2) are the possible values of ( c + d…

Let $\displaystyle \overrightarrow{\mathrm{c}}$ and $\displaystyle \overrightarrow{\mathrm{d}}$ be vectors such that $\displaystyle |\overrightarrow{\mathrm{c}}+\overrightarrow{\mathrm{d}}|=\sqrt{29}$ and $\displaystyle \overrightarrow{\mathrm{c}} \times(2 \hat{i}+3 \hat{j}+4 \hat{k})=(2 \hat{i}+3 \hat{j}+4 \hat{k}) \times \overrightarrow{\mathrm{d}}$. If $\displaystyle \lambda_1, \lambda_2\left(\lambda_1>\lambda_2\right)$ are the possible values of $\displaystyle (\vec{c}+\vec{d}) \cdot(-7 \hat{i}+2 \hat{j}+3 \hat{k})$, then the equation $\displaystyle \mathrm{K}^2 x^2+\left(\mathrm{K}^2-5 \mathrm{~K}+\lambda_1\right) x y+\left(3 \mathrm{~K}+\frac{\lambda_2}{2}\right) y^2-8 x+12 y+\lambda_2=0$ represents a circle, for K equal to :