Answer to Question #105714 in Geometry for Akanksha

Question #105714

Normals at any point P of ellipsoid x2/9+y2/4+z2=1 meet planes in Q1,Q2,Q3 resp. Show PQ1:PQ2:PQ3=9:4:1


1
Expert's answer
2020-03-19T12:59:36-0400

Equation of normal to the ellipse ax2+by2+cz2=1ax^2+by^2+cz^2=1 is xαaα=yβbβ=zγcγ\frac{x-\alpha}{a\alpha}=\frac{y-\beta}{b\beta}=\frac{z-\gamma}{c\gamma}

For the given elllipse a=19;b=14;c=1a=\frac{1}{9} ; b=\frac{1}{4} ; c=1

xxoxo/9=yyoyo/4=zzozo\frac{x-x_o}{x_o/9}=\frac{y-y_o}{y_o/4}=\frac{z-z_o}{z_o}

When x=0;y=5yo4;z=8zox=0;y=\frac{-5y_o}{4};z=-8z_o

Q1=(0,5yo4,8zo)Q_1=(0,\frac{-5y_o}{4},-8z_o)

When z=0;x=8xo/9;y=3yo/4z=0;x=8x_o/9;y=3y_o/4

Q3=(8xo/9,3yo/4,0)Q_3=(8x_o/9,3y_o/4,0)

When y=0;x=5xo9;z=3zoy=0; x=\frac{5x_o}{9};z=-3z_o

Q2=(5xo9,0,3zo)Q_2=(\frac{5x_o}{9},0,-3z_o)

P=(xo,yo,zo)P=(x_o,y_o,z_o)

PQ1=(xo0)2+(yo+5yo4)2+(zo+8zo)2PQ_1=\sqrt{(x_o-0)^2+(y_o+\frac{5y_o}{4})^2+(z_o+8z_o)^2} PQ1=(xo)2+(9yo4)2+(9zo)2PQ_1=\sqrt{(x_o)^2+(\frac{9y_o}{4})^2+(9z_o)^2}

PQ1=16xo2+81yo2+362zo2/4PQ_1=\sqrt{16x_o^2+81y_o^2+36^2z_o^2}/4


PQ3=(xo8xo/9)2+(yo3yo/4)2+(zo0)2PQ_3=\sqrt{(x_o-8x_o/9)^2+(y_o-3y_o/4)^2+(z_o-0)^2} PQ3=16xo2+81yo2+362zo2/36PQ_3=\sqrt{16x_o^2+81y_o^2+36^2z_o^2}/36


PQ2=(xo5xo9)2+(yo0)2+(zo+3zo)2PQ_2=\sqrt{(x_o-\frac{5x_o}{9})^2+(y_o-0)^2+(z_o+3z_o)^2} PQ2=(4xo9)2+(yo)2+(4zo)2PQ_2=\sqrt{(\frac{4x_o}{9})^2+(y_o)^2+(4z_o)^2}

PQ2=16xo2+81yo2+362zo2/9PQ_2=\sqrt{16x_o^2+81y_o^2+36^2z_o^2}/9

PQ1:PQ2:PQ3=1/4:1/9:1/36PQ_1:PQ_2:PQ_3=1/4:1/9:1/36

PQ1:PQ2:PQ3=9:4:1PQ_1:PQ_2:PQ_3=9:4:1



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