- #1

peroAlex

- 35

- 4

I'm a student of electrical engineering. At my math class, we were given a problem to solve at home. Now, from what I've managed to gather, this is a trick question, but I would like to get someone else's opinion on the task. It's also worth mentioning that parametrization is a relatively new concept as it was only yesterday we were dealing with it.

**1. The problem statement, all variables, and given/known data**

Parametrize the curve of intersection of ## x^2 + y^2 + z^2 = 1 ## and ## x - y = 0 ##.

## Homework Equations

Pardon me, but I was unable to collect "relevant equations" in this section.

## The Attempt at a Solution

##x^2 + y^2 + z^2 =1 ## represents a sphere with radius 1, while ## y = x ## represents a line parallel to x-axis. If I equate both sides and express for ##z## I acquire $$ z=-\sqrt{x^2-x+y^2+y-1} $$

I've plotted this function using Mathematica and it shows a plane. From this, I concluded that the intersection isn't a curve but a plane and cannot be parametrized.

I would like to ask for advice and maybe point me towards a solution. It seems a bit too trivial to hold.

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I also tried plugging ##x = y## into sphere equation and simplifying for ##y##. This returned a curve, "upside-down" parabola with the equation $$ y = \sqrt{\frac{1-x^2}{2}}$$

Could it be that parametrization follows from this procedure instead of the upper one?

I would like to thank in advance for your time and effort. Hope you're having a fantastic Saturday ;)