Straight Lines
Angle Bisector of Angle ABC — α²+β²
nta_pyq_2026_jan
Grade 11
Question:
Let $A(1,0)$, $B(2,-1)$ and $C\left(\dfrac{7}{3},\dfrac{4}{3}\right)$ be three points. If the equation of the bisector of the angle $ABC$ is $\alpha x+\beta y=5$, then the value of $\alpha^2+\beta^2$ is
Step-by-Step Solution
Key Concept: Unit vector along $BA$: $\vec{BA}=(-1,1)$, unit $=(-1/\sqrt{2},1/\sqrt{2})$. $\vec{BC}=(1/3,7/3)$, magnitude $=5\sqrt{2}/3$, unit $=(1/(5\sqrt{2}),7/(5\sqrt{2}))$.
$\alpha=3$, $\beta=1$. $\alpha^2+\beta^2=10$.
Correct Answer: 2