Straight Lines
Isosceles Triangle — Intersection Point and β⁴/α²
nta_pyq_2024_jan
Grade 11
Question:
Let $ABC$ be an isosceles triangle in which $A$ is at $(-1,0)$, $\angle A=\dfrac{2\pi}{3}$, $AB=AC$ and $B$ is on the positive $x$-axis. If $BC=4\sqrt{3}$ and the line $BC$ intersects the line $y=x+3$ at $(\alpha,\beta)$, then $\dfrac{\beta^4}{\alpha^2}$ is:
Step-by-Step Solution
Key Concept: With $A=(-1,0)$, $\angle A=120°$, isosceles. $B$ is on positive $x$-axis. By sine rule, find $B=(3,0)$. Line $BC$ has slope $-1/\sqrt3$ through $B=(3,0)$: $\sqrt3y+x=3$. Intersect with $y=x+3$.
$\beta^4/\alpha^2=36$.
Correct Answer: 36