Vector Algebra
Arc Midpoint — Equation with α and β
nta_pyq_2023_apr
Grade 12
Question:
Arc $PQ$ subtends right angle at centre $O$. Midpoint $R$ of arc. $\overrightarrow{OP}=\vec{u}$, $\overrightarrow{OR}=\vec{v}$, $\overrightarrow{OQ}=\alpha\vec{u}+\beta\vec{v}$. Then $\alpha,\beta^2$ satisfy
$x^2+x-2=0$
$x^2-x-2=0$
$x^2+x+1=0$
$x^2-x+1=0$
Step-by-Step Solution
Key Concept: $\overrightarrow{OQ}\perp\overrightarrow{OP}$ since right angle. $|\overrightarrow{OQ}|=r$. $\overrightarrow{OR}$ bisects angle $\Rightarrow$ angle $45°$.
$x^2-x-2=0$.
Correct Answer: 2