請問這幾題微積分怎麼算?

Question

1. find the shortest distance from the point (2,0,-3) to the plane x+y+z=1

2. the plane x+y+2z=2 intersects the paraboloid z=x^2+y^2 in an ellipse.Find the points on the ellipse that are nearest to and farthest from the origion.

Answer 1

最佳解答

1. find the shortest distance from the point (2,0,-3) to the plane x+y+z=1

Sol

設(2，0，－3)在x+y+z=1的垂足為(a，b，c)

a+b+c=1

(a－2)：(b－0)：(c+3)=1：1：1

(a－2)/1=b/1=(c+3)/1=(a－2+b+c+3)/(1+1+1)=(a+b+c+1)/3=2/3

d^2=(a－2)^2+b^2+(c+3)^2=3*(2/3)^2=4/3=12/9

d=2√3/3

2. the plane x+y+2z=2 intersects the paraboloid z=x^2+y^2 in an ellipse.Find the points on the ellipse that are nearest to and farthest from the origion.

Sol

2z=2－x－y=2x^2+2y^2

2x^2+x+2y^2+y=2

x^2+x/2+y^2+y/2=1

(x+1/4)^2+(y+1/4)^2=9/8

圓心(－1/4，－1/4),半徑=√(9/8)

直線方程式：x－y=0

(x+1/4)^2+(x+1/4)^2=9/8

(x+1/4)=9/16

x+1/4=+/-3/4

(1) x+1/4=3/4

x=1/2．y=1/2．z=1/2

(1/2．1/2．1/2)為最近點

(2) x+1/4=-3/4

x=－1，y=－1，z=2

(－1．－1．2)為最遠點