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 "ax^2+by^2+cz^2=1" is "\\frac{x-\\alpha}{a\\alpha}=\\frac{y-\\beta}{b\\beta}=\\frac{z-\\gamma}{c\\gamma}"

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

"\\frac{x-x_o}{x_o\/9}=\\frac{y-y_o}{y_o\/4}=\\frac{z-z_o}{z_o}"

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

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

When "z=0;x=8x_o\/9;y=3y_o\/4"

"Q_3=(8x_o\/9,3y_o\/4,0)"

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

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

"P=(x_o,y_o,z_o)"

"PQ_1=\\sqrt{(x_o-0)^2+(y_o+\\frac{5y_o}{4})^2+(z_o+8z_o)^2}" "PQ_1=\\sqrt{(x_o)^2+(\\frac{9y_o}{4})^2+(9z_o)^2}"

"PQ_1=\\sqrt{16x_o^2+81y_o^2+36^2z_o^2}\/4"


"PQ_3=\\sqrt{(x_o-8x_o\/9)^2+(y_o-3y_o\/4)^2+(z_o-0)^2}" "PQ_3=\\sqrt{16x_o^2+81y_o^2+36^2z_o^2}\/36"


"PQ_2=\\sqrt{(x_o-\\frac{5x_o}{9})^2+(y_o-0)^2+(z_o+3z_o)^2}" "PQ_2=\\sqrt{(\\frac{4x_o}{9})^2+(y_o)^2+(4z_o)^2}"

"PQ_2=\\sqrt{16x_o^2+81y_o^2+36^2z_o^2}\/9"

"PQ_1:PQ_2:PQ_3=1\/4:1\/9:1\/36"

"PQ_1:PQ_2:PQ_3=9:4:1"



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