Coordinate Geometry
Minimum distance expression — integer answer
MJAT_TS5_P2
Grade 12
Question:
The minimum value of $\left[(t-r)^2+\left(\sqrt{12-1-t^2}-4r\right)^2\right]^{1/2}$ for all permissible values of $t$ and $r$ is equal to $a-\sqrt{b}\cdot\sqrt{c}$ where $a,b,c\in\mathbb{N}$ ($b$ is prime). Then $a+b-c-4=$
Step-by-Step Solution
Key Concept: The expression is the distance between point $(t, \sqrt{11-t^2})$ on a circle (or arc) and point $(r, 4r)$ on a line. The minimum distance is the shortest distance from the circle to the line.
$a+b-c-4=\mathbf{6}$.
Correct Answer: 6