Consider the parabaloid z = x 2 + y 2 . The plane 6 x ? 5 y + z ? 3 = 0 cuts the

Question

Consider the parabaloid z = x2 +y2. The plane 6x?5y+z?3 = 0 cuts the parabaloid, its intersection being a curve. Find a parametriza-tion of this curve by doing the following:

(i) Find the intersection of the parabaloid and the plane in terms of x and y (that is, its projection onto the xy-plane).

(ii) This projection is a circle, express the circle in terms of (x ? a)2 + (y ? b)2 = r2 by completing the square.

(iii) Parametrize this circle. (Hint: Recall that to parametrize x2 + y2 =r2,wesetx=rcost,y=rsintwhere0?t<2?.Use this to parametrize your circle)

(iv) Using the parametrization in part (iii), give a parametrization of the curve given as the intersection of the parabaloid and plane

Explanation / Answer

(i)

6x-5y +x^2 +y^2 -3 = 0

(ii)

(x+3)^2 +(y-2.5)^2 = 18.25

(iii)

x = -3+4.27 cost

y = 2.5 + 4.27 sint

Related Questions

Navigate

• Browse (All)
• Browse C
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.