The answer is 5.44.

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hey what was the discussion going on in fiitjee beside sb 2when we were about to leave

Kaustubh Miglani
- 5 years, 8 months ago

lakshay sinha,dont u dare ignore my sir

tomi dfvghj
- 5 years, 8 months ago

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i am not a teacher tomi.i am just a student struggling to improve his skills

Kaustubh Miglani
- 5 years, 8 months ago

What do you mean?

Department 8
- 5 years, 8 months ago

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he was referring to me.he believes i am his teacher.and well for you did not respond he was too angry

Kaustubh Miglani
- 5 years, 8 months ago

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@Kaustubh Miglani – Give me the link for the respondance.

Department 8
- 5 years, 8 months ago

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@Department 8 – link to respondance means

Kaustubh Miglani
- 5 years, 8 months ago

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@Kaustubh Miglani – For what he was asking.

Department 8
- 5 years, 8 months ago

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@Department 8 – he was asking for a reply to comment dated 4 days 3 hrs ago on this page

Kaustubh Miglani
- 5 years, 8 months ago

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@Kaustubh Miglani – Sorry did not saw that. We were shifting to another place and today my internet was restored.

Department 8
- 5 years, 8 months ago

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@Department 8 – no prob buddy gonna come tomorrow

Kaustubh Miglani
- 5 years, 8 months ago

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@Kaustubh Miglani – Yep what is tomorrow's time table? Bio& Maths?

Department 8
- 5 years, 8 months ago

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@Department 8 – maths and sst listen to this theme u would be motivated https://www.youtube.com/watch?v=nemTwRw6Am8

Kaustubh Miglani
- 5 years, 8 months ago

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@Kaustubh Miglani – Dude Speaker's not working right now, you should listen to Fire Inside By Gemini.

Department 8
- 5 years, 8 months ago

@Department 8 – well lakshay do u have any idea if pacific mall is gonna be open on 2nd october

Kaustubh Miglani
- 5 years, 8 months ago

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@Kaustubh Miglani – Yeah, google says yes. (Bookmyshow.com reference)

Department 8
- 5 years, 8 months ago

Nice solution. In which class you are? Are you a fiitjee student?

Priyanshu Mishra
- 5 years, 7 months ago

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he is in 9th along with me and yes he is a fiitjee student

Kaustubh Miglani
- 5 years, 7 months ago

To complete your solution, you must also check that equality can occur. In other words, you must show that there actually exist $a$ , $b$ , $c$ , $d$ such that $b^2 + d^2 = 49/9$ . Here, the values that work are $a = \frac{27}{\sqrt{130}}, \ b = c = \frac{21}{\sqrt{130}}, \ d = \frac{49}{3 \sqrt{130}}.$

Jon Haussmann
- 5 years, 7 months ago

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By Cauchy-Schwarz Inequality we have

$\large{{ \left( { a }^{ 2 }+{ c }^{ 2 } \right) \left( { b }^{ 2 }+{ d }^{ 2 } \right) }\ge { \left( ab+cd \right) }^{ 2 }}$

We will go back to the basics of Chebysev's Identities which stated that if $a_{1}\ge a_{2} \ge a_{3}........a_{n-1} \ge a_{n}$ and $b_{1} \ge b_{2} \ge b_{3}.........b_{n-1}\ge b_{n}$ .

$\large{{ \frac { \sum _{ i=1 }^{ n }{ \left( { a }_{ i } \right) \left( { b }_{ i } \right) } }{ n } }\ge \frac { \prod _{ i=1 }^{ n }{ { a }_{ i } } }{ n } \times \frac { \prod _{ i=1 }^{ n }{ { b }_{ i } } }{ n } \ge \frac { \sum _{ i=1 }^{ n }{ \left( { a }_{ i } \right) \left( { b }_{ n+1-i } \right) } }{ n } }$

In the question we are given $a \ge d, b\ge c$ . Applying Cheybsev's Identity

$\large{\left( ab+cd \right) \ge \left( ac+bd \right) \\ \left( ab+cd \right) \ge 7\\ { \left( ab+cd \right) }^{ 2 }\ge 49}$

And from the first equation we have

$\large{{ \left( { a }^{ 2 }+{ c }^{ 2 } \right) \left( { b }^{ 2 }+{ d }^{ 2 } \right) }\ge { \left( ab+cd \right) }^{ 2 }\\ \left( { a }^{ 2 }+{ c }^{ 2 } \right) \left( { b }^{ 2 }+{ d }^{ 2 } \right) \ge 49\\ \left( { b }^{ 2 }+{ d }^{ 2 } \right) \ge \frac { 49 }{ 9 } =5.44}$

equality holds when $\large{a=\dfrac{27}{\sqrt{130}}, b=c=\dfrac{21}{\sqrt{130}}, d=\dfrac{49}{3\sqrt{130}} }$