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Question
I am having difficulty with this problem.

Consider the sphere S of radius 5 centred at (1, 0,2). Find the equation of the plane tangent to S at the point (5,3,2).

Hint: To get started, draw this situation in 2D with a circle.
I am having difficulty with this problem.

Answer
Since the sphere and the circles center are both a z=2, the third dimension can be dropped for a moment.  Find the equation of a tangent line to the circle (x-1)^2 + y^2 = 5^2 at (5,3).

To find a perpendicular to the line going from (1,0) to (5,3), note that the size is (5-1,3-0) =
(4,3).  Flip the x and y values and negate one of them.  This gives (3,-4) or (-3,4).  Note that one of those is the negative of the other.  This is done because the dot product of (a,b)(-b,a) is -ab + ab = 0.  This means the lines are perpendicular since the dot product is 0.

The equation of the line would be 3x - 4y = 3*5 - 4*3 = 3.
That is, it is 3x - 4y = 3.

Getting back to 3 dimensional, let z be any value with 3x - 4y = 3.
This will generate a plane that is straight up and down crossing through the horizontal x-y plane.

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I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can also say that I broke 5 minutes for a mile, which is over 12 mph, but is that relevant?

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Experience in the area; I have tutored people in the above areas of mathematics for over two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree back in 1984. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

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My master's paper was published in the OSU journal. The subject of it was Numerical Analysis used in shock waves and rarefaction fans. It dealt with discontinuities that arose over time. They were solved using the Leap Frog method. That method was used and improvements of it were shown. The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.

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Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

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I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

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