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Find the Length of a Curve

Suppose we have the following curves:

$C_1$: $y^2=\frac{1}{2}x^2+a^2$
$C_2$: $x^2+y^2=a^2$
$C_3$: $x^2+y^2=a^2+\frac{1}{2}x^2$

I need to find the length of the curves and show that:
$$A\le B\le C_2\le C_3$$
I know that the length of $C_3$ is simply $2a$.
Do I need to find the length of the curves first and then try to show that or do I need to find the length first? Which method is faster?

A:

This is all obvious geometry, so you don’t have much to do here.
The length of $C_1$ is the distance from the point $(-a,0)$ to the point $(a,0)$ along the $y$-axis, so it’s $\sqrt{a^2+a^2} = a$, and the length of $C_2$ is the distance from the point $(0,a)$ to the point $(0,-a)$ along the $x$-axis, so it’s $\sqrt{a^2+a^2} = 2a$.
We also know the length of $C_3$ is $2a$, and you do the rest.

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