Ellipsoid
Ellipsoid
The equation of an ellipsoid is given by:
\[\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{c^{2}}=1\]PYQs
Ellipsoid
1) Find the length of the normal chord through a point \(P\) of the ellipsoid
\[\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1\]and prove that if it is equal to \(4PG_3\), where \(G_3\) is the point where the normal chord through \(P\) meets the \(xy-plane\), then \(P\) lies on the cone
\[\dfrac{x^6}{a^6}(2c^2-a^2)+\dfrac{y^6}{b^6}(2c^2-b^2)+\dfrac{z^4}{c^4}=0\][2019, 15M]
2) Three points \(P\), \(Q\) and \(R\) are taken on the ellipsoid \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{c^{2}}=1\) so that lines joining to \(P\), \(Q\) and \(R\) to origin are mutually perpendicular. Prove that plane \(P Q R\) touches a fixed sphere.
[2011, 20M]
3) Show that the enveloping cylinders of the ellipsoid \(a^{2} x^{2}+c^{2} z^{2}=1\) 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 \(3 x+2 y-z+2=0\) at the point \((1,-2,1)\) and cuts orthogonally the sphere \(x^{2}+y^{2}+z^{2}-4 x+6 y+4=0\).
[2006, 15M]
5) If the plane \(l x+m y+n z=p\) passes through the extremities of three conjugate semi-diameters of the ellipsoid \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{c^{2}}=1\), prove that \(a^{2} l^{2}+b^{2} m^{2}+c^{2} n^{2}=3 p^{2}\).
[2006, 15M]
6) If normals at the points of an ellipse whose eccentric angles are \(\alpha\), \(\beta\), \($\gamma$ and\)\delta$$ meet in a point, then show that
\[\sin(\beta+\gamma)+ \sin(\gamma + \alpha)+ \sin(\alpha+\beta)=0\][2005, 12M]
7) Prove that:
\[\int \dfrac{x^2+y^2}{p} dx = \dfrac{\pi a b}{4} [4+(a^2+b^2)(a^-2+b^-2)],\]when the integral is taken round the ellipse \(\dfrac{x^2}{a^2}+\dfrac{y^2}{a^2}=1\) and \(p\) is the length of the perpendicular from the centre to the tangent.
[2004, 15M]
8) Tangent planes are drawn to the ellipsoid \(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} + \dfrac{z^2}{c^2}=1\) through the point \((\alpha, \beta, \gamma)\). Prove that the perpendiculars to them through the origin generate the cone \((\alpha x + \beta y + \gamma z)^2\) = \(a^2x^2 + b^2y^2+c^2z^2\).
[2004, 15M]
9) Find the locus of equal conjugate diameters of the ellipsoid \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{c^{2}}=1\).
[2001, 15M]