Doubt

1) How to prove this one: $\frac{2}{a+b} + \frac{2}{b+c} + \frac{2}{c+a}$ $\le$ $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$

2) Eliminate t: $x=10( t-\sin t)$ and $y=10(1-\cos t)$

Easy Math Editor

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Comments

This is a equation of a cycloid . I have eliminated $t$ but equation is Messier. Generally , $x=r(t-\sin(t))$

,and $y=r(1-\cos(t))\implies \cos(t)= 1-\dfrac{y}{r} \implies \sqrt{1-\sin^2(t)} = 1-\dfrac{y}{r} \rightarrow{a}$

And also $t=\cos^{-1} (1-\dfrac{y}{r})$

From $x$ we can find $\sin(t)= t-\dfrac{x}{r} = \cos^{-1}(1-\dfrac{y}{r})-\dfrac{x}{r}$

Putting this on $a$ we will find equation is $\dfrac{x^2+y^2}{r^2} -\frac{2}{r}(x \cos^{-1}(\dfrac{y}{r}-1)+y) +(\cos^{-1} (\dfrac{y}{r}-1))^2 =0$

Dwaipayan Shikari - 2 months, 3 weeks ago

Please note that I am actually trying to derive the equation of the path traced by a point fixed on a rolling body in pure rolling in space. Please help, I am a novice.

Baibhab Chakraborty - 2 months, 3 weeks ago

Ok thank you sir so much.

Baibhab Chakraborty - 2 months, 3 weeks ago

Thanks bro . But I am not a sir!;)

Dwaipayan Shikari - 2 months, 3 weeks ago

A Former Brilliant Member - 2 months, 2 weeks ago

Is there any available facility of private chat in brilliant? Just intrigued.

Baibhab Chakraborty - 2 months, 2 weeks ago

No, there is no such feature (tho there should be lol)

A Former Brilliant Member - 2 months, 2 weeks ago

@Percy Jackson I found a solution to the first one: I had thought of using H.M but couldn;t end doing the problem. Its as follows:

$\frac{a+b}{2} \ge \frac{2ab}{a+b}$ [ A.M $\ge$ H.M ]

$\frac{a+b}{2ab} \ge \frac{2}{a+b}$

$\frac{\frac{1}{a} + \frac{1}{b}}{2} \ge \frac{2}{a+b}$

$\Sigma \frac{\frac{1}{a}+\frac{1}{b}}{2} \ge \Sigma \frac{2}{a+b}$

$\frac{2}{a+b} + \frac{2}{b+c} + \frac{2}{c+a} \le \frac{1}{a} + \frac{1}{b} + \frac{1}{c}$

Baibhab Chakraborty - 2 months, 1 week ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$