Vector Algebra
Vectors
star_batch_jee_advanced_2025
Grade 12
Question:
If $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ are on a circle of radius $R$ whose centre is at origin and $\vec{c} - \vec{a}$ is perpendicular to $\vec{d} - \vec{b}$, then $|\vec{d} - \vec{a}|^2 + |\vec{b} - \vec{c}|^2$ (AC is diameter)
Step-by-Step Solution
Key Concept: Vector magnitude expansion combined with geometric angle relationships yields the final formula.
Expanding $|\vec{d} - \vec{a}|^2 + |\vec{b} - \vec{c}|^2$ and simplifying yields $4R^2 - 2(\vec{d} - \vec{a}) \cdot (\vec{b} - \vec{c}) = 4R^2 - 2R^2(\cos \angle AOB + \cos \angle BOC) = 4R^2(1 - \cos \angle AOB \cos \angle BOC)$ where $\angle AOB = \pi - \angle BOC$.
Correct Answer: 4