Coordinate Geometry
Points on y = 1/x — angle relationship via midpoint
MJAT_TS4_P1
Grade 12
Question:
Consider two points $A(x_1,y_1)$ and $B(x_2,y_2)$ on the graph of $y=\dfrac{1}{x}$ such that $0<x_1<x_2$ and $OA\perp OB$ (where $O$ is the origin). Let $C$ be the midpoint of segment $AB$. Then:
A) The angle between the $x$-axis and ray $OA$ equals three times the angle between the $x$-axis and ray $OC$
B) The angle between the $x$-axis and ray $OA$ equals two times the angle between $x$-axis and ray $OC$
C) The angle between the $x$-axis and ray $OA$ equals the angle between $x$-axis and ray $OC$
D) The angle between the $x$-axis and ray $OA$ equals four times the angle between $x$-axis and ray $OC$
Step-by-Step Solution
Key Concept: Write $A=(t, 1/t)$ and $B=(s, 1/s)$. $OA\perp OB$ means $\vec{OA}\cdot\vec{OB}=0\Rightarrow ts+\frac{1}{ts}=0$ (impossible for real positive $t,s$... wait: $OA\perp OB$ means direction vectors are perpendicular). Angle of $OA$ with $x$-axis: $\tan\alpha = 1/t^2$. Midpoint $C=\left(\frac{t+s}{2}, \frac{1/t+1/s}{2}\right)$.
$\angle AOx = 3\angle COx$. Answer: **A**.
Correct Answer: A