Coordinate Geometry: JEE Main 2026 Mathematics question

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