Question

Let P = (0,0, 2)and let S be the unit sphere with equation x2 + y2 + z2 = 1.The collection of points on the sphere where the tangent plane of the sphere contains the point Pforms a curve. Parametrize this curve.

Answer 1

2 2 2 5) f(x,y,z) » n'tý x² + y² + z2 = 1 Equation of sougent at Cry 31,7) fo=df - 2 = 2n ty=d7-24 12-04 22 dz dy dn fo(2,803,) (x-u.) + f (x,71,7) (Y-Y) + fez(X., Y., 2) (2-2,) - 0 z .. 2x, CX-X,) + 24, C9-yo) + 22, (2-4) 2. - ( • a-x1) + 0-2) + 2(z- z) ) a XX, - %;? + yoy, - y2 + 221 - 22 = 0. contains. (0, 0, 2) The tongent Plane contains. 1. (xxo) – x2 + Cy xo) - y2 + (2x2) –22 = 0 + o-4² + 22, 22-0. : Locus of the points on surfan s where the tangent plone of the spone contains Plo, 0,2g is. x² + y² & 2² 2 229 o

Since the points on surface S. . x² + y2 + z² = 1 x² + y² + z² -22=0 ľ 22 2 O. 2= 1/2 ņ Please opvote J.