Applications of Derivatives
Maxima and Minima — Distance Function
nta_pyq_2024_apr
Grade 12
Question:
Let the maximum and minimum values of $\left(\sqrt{8x-x^2-12}-4\right)^2+(x-7)^2$, $x\in\mathbb{R}$, be $M$ and $m$ respectively. Then $M^2-m^2$ is equal to _________.
Step-by-Step Solution
Key Concept: Let $y=\sqrt{8x-x^2-12}$: then $(x-4)^2+y^2=4$ (circle centre $(4,0)$, radius 2). Expression is squared distance from $(x,y)$ on circle to $(7,4)$.
$M=41$, $m=9$. $M^2-m^2=1600$.
Correct Answer: 1600