Ellipsoid
PYQs
Ellipsoid
1) Find the length of the normal chord through a point P of the ellipsoid
x2a2+y2b2+z2c2=1and prove that if it is equal to 4PG3, where G3 is the point where the normal chord through P meets the xy−plane, then P lies on the cone
x6a6(2c2−a2)+y6b6(2c2−b2)+z4c4=0[2019, 15M]
2) Three points P, Q and R are taken on the ellipsoid x2a2+y2b2+z2c2=1 so that lines joining to P, Q and R to origin are mutually perpendicular. Prove that plane PQR touches a fixed sphere.
[2011, 20M]
3) Show that the enveloping cylinders of the ellipsoid a2x2+c2z2=1 with generators perpendicular to z−axis meet the plane z=0 in parabolas.
[2008, 20M]
4) Find the equation of the sphere which touches the plane 3x+2y−z+2=0 at the point (1,−2,1) and cuts orthogonally the sphere x2+y2+z2−4x+6y+4=0.
[2006, 15M]
5) If the plane lx+my+nz=p passes through the extremities of three conjugate semi-diameters of the ellipsoid x2a2+y2b2+z2c2=1, prove that a2l2+b2m2+c2n2=3p2.
[2006, 15M]
6) If normals at the points of an ellipse whose eccentric angles are α, β, $γ$and\delta$$ meet in a point, then show that
sin(β+γ)+sin(γ+α)+sin(α+β)=0[2005, 12M]
7) Prove that:
∫x2+y2pdx=πab4[4+(a2+b2)(a−2+b−2)],when the integral is taken round the ellipse x2a2+y2a2=1 and p is the length of the perpendicular from the centre to the tangent.
[2004, 15M]
8) Tangent planes are drawn to the ellipsoid x2a2+y2b2+z2c2=1 through the point (α,β,γ). Prove that the perpendiculars to them through the origin generate the cone (αx+βy+γz)2 = a2x2+b2y2+c2z2.
[2004, 15M]
9) Find the locus of equal conjugate diameters of the ellipsoid x2a2+y2b2+z2c2=1.
[2001, 15M]