(1) Find parametric equations for the line through the point (0, 1,2) that is parallele to the plane x+y+z = 2 and perpendicular to the line x = 1 +t, y = 1 ?t,z = 2t.
(2) An ellipsoid is created by rotating the ellipse 4x2 + y2 = 16 about the x-axis. Find an equation of the ellipsoid.

The solutions are explained in the attached Word file, in red.
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Solutions:

(1) We note that
(a) A line parallel to plane x + y + z = ...

Solution Summary

Parametric Equations and the Equation of an Ellipsoid are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

... The parametric equations are ... The equations are ... z2 y2 1 (2.5) 9 This is the canonical equation of an ellipse, centered at (0,0,0) with a major axis a 3 ...

... 15. Calculate sin 87° accurate to five decimal places using Taylor's formula for an ... 4. Write the equation of the line tangent to the parametric curve x ...

... For instance, for the parametric curve r(t) = (cos t, sin t, 0 ... say f (x, y ) = 0 and the region D is an ellipse. To obtain D set z = 0 in the equation x2 + 2y 2 ...

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... The arc length for a curve represented by the parametric equation: ... Dividing equations (1)-(3) by 2, squaring and plugging into equation (4) gives: ...

... y − 1)2 ( x + 3) 2 =1 + 22 ( 3) 2 So the ellipse equation is ( x + 3) 2 ( y − 1) 2 =1 + 3 4 8. Graph by hand the curve whose parametric equations are given ...

... we get (y + 1)2 (x − 2)2 + =1. 9 x2 y2 + = 1 , we see that the given parametric Comparing it with the standard equation a2 b2 equations represent an ellipse. ...

... of parametric equations for the rectangular equation: y = 2x - 2. x = t/2; y = t - 1. y = 2t; 2x = t + 2. y = 2t2 - 2. x = t; y = 2t - 2. 18 of 25. Graph the ellipse...

... ii) at the points P and Q. Show that the tangents at P and Q to ellipse (ii) are at right angles to one another. Please show this using parametric equations. ...