Least value -1

Calculus Level 3

If $a$ is the least value of

$\sqrt{(u-v)^2+(\frac{9}{u}-\sqrt{4-v^2})^2},$

then what is the value of $\lfloor a \rfloor?$

Note: $u,v \in \mathbb{R}.$

The answer is 2.

1 solution

Ronak Agarwal
Nov 2, 2014

It is clear by inspection that the given expression is the distance between any point on the hyperbola $xy=9$ and a point on the circle ${x}^{2}+{y}^{2}=4$ .

Anything

By drawing their graphs on it is very easy to observe that the minimum distance occurs between the points $(\sqrt{2},\sqrt{2})$ and $(3,3)$ . hence the minimum distance is given by :

Min(distance)=Min(expression)= $\sqrt{2}(3-\sqrt{2})=3\sqrt{2}-2$

Do you know what is the mathematical condition to find the minimum distance between 2 (differentiable) curves?

Eyepower 2000 isn't a valid answer, as that heavily depends on the accuracy of your graph.

Calvin Lin Staff - 6 years, 7 months ago

In this question too I have used the fact that minimum distance between two curves is the distance between points whose tangents are parellel.

Ronak Agarwal - 6 years, 7 months ago

Right. The idea is that these points share the common normal line, which is why the distance is minimal / maximal.

Calvin Lin Staff - 6 years, 7 months ago

If in a question drawing a graph is not viable then I would be finding the distance between the points whose tangents are parellel.

IThis thing often solves many questions like this.

Ronak Agarwal - 6 years, 7 months ago

Yeah, its the elegant and the best approach to solve this problem. Keep it up !

Sandeep Bhardwaj - 6 years, 7 months ago

Really a nice way....

Sanjeet Raria - 6 years, 7 months ago

Least value -1

The answer is 2.

1 solution

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