Coordinate Geometry
Triangle with variable third side; interior point condition
Grade Class 12
Question:
Two sides of a triangle have combined equation $x^2-3y^2-2xy+8y-4=0$. The third side passes through $(-5,1)$ and the origin is interior. If the range of slope of the third line is $(a,b)$, then $\dfrac{a+\frac{1}{b^2}}{4}$ is
Step-by-Step Solution
Key Concept: Factor: $x^2-3y^2-2xy+8y-4=(x-3y+2)(x+y-2)=0$. Lines: $x-3y+2=0$ (slope $1/3$) and $x+y-2=0$ (slope $-1$). For origin to be interior, find valid slope range of third side.
$\frac{a+1/b^2}{4}=6$.
Correct Answer: 6