炘涛 發問時間: 科學數學 · 5 年前

請問這幾題微積分怎麼算?

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.

1 個解答

評分
  • 5 年前
    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

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