00:00The arithmetic mean of numbers x and y is 5, and their geometric mean is 4.
00:03The question asks what the value of the expression 1 divided by x squared plus 1 divided by y squared is, if the problem is far away.
00:08First of all, the arithmetic mean of x and y is x plus y divided by 2 equals 5, right?
00:14So what do we find the sum of x plus y to be?
00:17We find it to be 10.
00:19So when we look at the geometric mean, why the square root of x times y?
00:23Since there are two numbers, that equals 4, right?
00:26Yes.
00:26If we square both sides, why, friends?
00:30Because we'll need it now.
00:32x times y equals 16, so we stop there.
00:35Since it was calculated square by square, it came out as x times y.
00:37So, friends, what is expected of us?
00:401 divided by x squared plus 1 divided by y squared.
00:44If we divide this a bit, y squared and x squared, y squared is x squared plus y squared divided by x squared multiplied by...
00:53We find it as y squared.
00:54So, what exactly is x squared plus y squared, guys?
00:56It was the sum of the squares of the two numbers.
00:58How were we able to write this?
01:00We can write it as x plus y squared minus 2xy divided by x squared multiplied by y squared, right guys?
01:07Yes.
01:08So, what is the square of x plus y, guys?
01:11We'll write it as its square.
01:13That's 100 too.
01:14What is minus 2xy, guys?
01:16Minus 2 times 16.
01:19What is x squared y squared equal to, friends?
01:22Of course, 256.
01:27When we do that, friends, if you subtract 32 from 100, how much is left?
01:33Of course it's 68.
01:37256.
01:40That's also equal to that, friends.
01:41Let's switch everything to the right now.
01:43Can these be divided by 4?
01:45It divides.
01:50We find our answer as 17 divided by 64.
01:54We think this would be the answer.
01:55Thank you for watching.
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