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Numerade Educator

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Problem 20 Easy Difficulty

Show that the equation represents a sphere, and find its center and radius.

$ 3x^2 + 3y^2 + 3z^2 = 10 + 6y + 12z $

Answer

center $(0,1,2), \quad r=\frac{5 \sqrt{3}}{3}$

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Video Transcript

this is a question in which we need to show that they personally presented sphere and we need to find if center and radius Question is three x esquire place the U. S. Square. Yes, freezer square equal to 10 plus six way Yes. General said Let us rewrite as three x esquire The three Wise Square. The trees are square minus six way minus 12. Said equal to 10. So If you divide both sides by three we will be getting access choir plus y squared less than a square -2 Y -4. said Equal to 10 x three. So this is definitely in the form of access squared plus y squared. Access squad plus y squared plus Z squad. Bless two attacks two attacks plus two y plus two. These are equal to D or r. Okay. Oh that's right. The this is in the form of here, this is the general farmer. It's clear. And anyhow if we have ah that uh what we said coefficient of access square by square and that is quite equal the same. Equal to the same. And we if we do not have any terms like x, Y, why is that? And said X. So this definitely represents there is fear now center center will be my energy minus. I'm sorry center will be minus action minus of minus g minus at minus f. My energy. So if you compare to act there is no term containing X. So this will be zero and we have f. So two f is -2, That is the coefficient of Y. So F will be -1 and this will be simply one. Similarly, she will be minus to the minus, It will be this is the center and radius will be equal to at a square block. G square plus f square minus D under the square root, so zero plus one square AR -1 is well we should say Oh Jeez -1. Less -2 Squared and minus 10 by three. Under the square root, one plus four minus 10 by three. This is 15 -10, so five by three under the schedule, So radius will be equal to fight by 300 square root. So if we want to rationalize it will become 15 under 15 by And the road 15 x three. This is the areas, this is that centre. Thank you.