We will cover following topics

Ellipsoid

The equation of an ellipsoid is given by:

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1

PYQs

Ellipsoid

1) Find the length of the normal chord through a point $P$ of the ellipsoid

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1

and prove that if it is equal to $4 P G_{3}$ , where $G_{3}$ is the point where the normal chord through $P$ meets the $x y - p l a n e$ , then $P$ lies on the cone

\frac{x^{6}}{a^{6}} (2 c^{2} - a^{2}) + \frac{y^{6}}{b^{6}} (2 c^{2} - b^{2}) + \frac{z^{4}}{c^{4}} = 0

[2019, 15M]

2) Three points $P$ , $Q$ and $R$ are taken on the ellipsoid $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1$ 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 $z - a x i s$ 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 $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1$ , 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 $α$ , $β$ , $$ γ $ a n d$ \delta$$ meet in a point, then show that

\sin (β + γ) + \sin (γ + α) + \sin (α + β) = 0

[2005, 12M]

7) Prove that:

\int \frac{x^{2} + y^{2}}{p} d x = \frac{π a b}{4} [4 + (a^{2} + b^{2}) (a^{-} 2 + b^{-} 2)],

when the integral is taken round the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{a^{2}} = 1$ and $p$ is the length of the perpendicular from the centre to the tangent.

[2004, 15M]

8) Tangent planes are drawn to the ellipsoid $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1$ through the point $(α, β, γ)$ . Prove that the perpendiculars to them through the origin generate the cone $(α x + β y + γ z)^{2}$ = $a^{2} x^{2} + b^{2} y^{2} + c^{2} z^{2}$ .

[2004, 15M]

9) Find the locus of equal conjugate diameters of the ellipsoid $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1$ .

[2001, 15M]

< Previous Next >