2004

1) Find the equation of the sphere for which the circle $x^2+y^2+z^2+2x-4y+5=0$, $x-2y+3z$ $+1=0$ is a great circle.

[10M]

2) A variable plane is at a constant distance $p$ from the origin and meets the axes at $A$, $B$ and $C$. Through $\mathrm{A},\mathrm{B}$ and $\mathrm{C}$, the planes are drawe parallel to the coordinate planes. Find the locus of their point of intersection.

[10M]

3) 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 $x^2+y^2+z^2-4x+6y+4-0$.

[10M]

4) Find the equation of the right circular cone generated by straight lines drawn from the origin to cut the circle throughthe three points $(1,2,2)$, $(2,1,-2)$ and $(2,-2,1)$.

[10M]

5) Find the equations of the tangent planes to the ellipsoid $7{\mathrm{x}}^2+5{\mathrm{y}}^2+3{\mathrm{z}}^2=60$ which pass through the line $7x+10y-30=0,5y-3z=0$.

[10M]

2003

1) A line makes angles $\alpha ,\beta ,\gamma ,\delta$ with the four diagonals of a cube. Show that

[10M]

2) Show that the equation $\sqrt{fx}+\sqrt{8y}+\sqrt{hz}=0$ represents a cone that fouches the co-ordinate planes and that the equation to its reciprocal cone is $fyz+gzx+hxy=0$.

[10M]

3) Show that any two generators belonging to the different system of generating lins of a hyperboloid of one sheet intersect.

[10M]

4) Show that the locus of a point from which three mutually perpendicular tangent lines can be drawn to the paraboloid

$a{x}^2+b{y}^2+2z=0$ is
$ab(x^2+y^2)+2(a+b)z=1$.

[10M]

5) Show that the enveloping cylinder of the ellipsoid ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}+{\displaystyle \frac{z^2}{c^2}}=1$ whose generators are parallel to the line ${\displaystyle \frac{x}{0}}={\displaystyle \frac{y}{\pm \sqrt{a^2-b^2}}}={\displaystyle \frac{z}{c}}$ meet the plane $\mathrm{z}=0$ in circles.

[10M]

2002

1) Show that any two circular sections of an ellipsoid of opposite systems lie on a sphere.

[10M]

2) Prove that the locus of the line of intersection of perpendicular tangent plancs to the cone

$a{x}^2+b{y}^2+c{z}^1=0$ is the cone $a(b+c)x^2+b(c+a)y^2+c(a+b)z^2=0$

TBC

[10M]

3) Prove that two normals to the ellipsoid ${\displaystyle \frac{x^2}{4}}+{\displaystyle \frac{y^2}{3}}+{\displaystyle \frac{z^2}{2}}=1$ lie in the plane $x+2y+3z=0,$ and the line joining their feet has direction cosines proportional to $12,-9,2$.

[10M]

4) Prove that the shortest distances between the diagonals of a rectangular parallelopiped and edges not meeting them are
${\displaystyle \frac{bc}{\sqrt{b^2+c^2}}}\cdot {\displaystyle \frac{ca}{\sqrt{c^2+a^2}}}$ and ${\displaystyle \frac{ab}{\sqrt{a^2+b^2}}}$ where $a$, $b$, $c$ are lengths of the edges of a parallelopiped.

[10M]

5) Find the equation of the enveloping cylinder of the sphere $x^2+y^2+z^2-2y-4z=1$ having its generalor parallel to the line $x=2y=2z$.

[10M]

2001

1) Prove that the polar of one limiting point of a coaxial system of circles with respect to any circle of the system passes through the other limiting point.

[10M]

2) $CP$ and $CD$ are conjugate diameters of an ellipse ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$. Prove that the locus of the orthocentre of the triangle $\mathrm{C}\mathrm{P}\mathrm{D}$ is the curve $2{(b^2{y}^2+a^2{x}^2)}^3={(a^2-b^2)}^2(b^2{y}^2-a^2{x}^2)$.

[13M]

3) If ${\displaystyle \frac{x}{1}}={\displaystyle \frac{y}{2}}={\displaystyle \frac{z}{3}}$ represents one of the that mutually perpendicular generators of the cone $5yz-8zx-3xy=0$. Find the equations of the other two.

[13M]

4) If the section of the enveloping cone of the ellipsoid ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}+{\displaystyle \frac{z^2}{c^2}}=1,$ whose vertex is $\mathrm{P},$ by the plane $\mathrm{z}=0$ is a rectanagular hyperbola, prove that the locus of $P$ is ${\displaystyle \frac{x^2+y^2}{a^2+b^2}}+{\displaystyle \frac{z^2}{c^2}}=1$

[14M]

2000

1) Sketch the conic $[xy1]\left[\begin{array}{ccc}5& 3& -5\\ 3& 5& -3\\ -5& -3& -3\end{array}\right]\left[\begin{array}{l}x\\ y\\ 1\end{array}\right]=0$

[10M]

2) A plane passes through a fixed point $(2p,2q,2r)$ and cuts the axes in $\mathrm{A},\mathrm{B},\mathrm{C}$. Show that the locus of the centre of the sphere $\mathrm{O}\mathrm{A}\mathrm{B}\mathrm{C}$ is ${\displaystyle \frac{p}{x}}+{\displaystyle \frac{q}{y}}+{\displaystyle \frac{r}{z}}=1$.

[10M]

3) A straight line $\mathrm{A}\mathrm{B}$ of fixed length moves so that its extremities $\mathrm{A},\mathrm{B}$ lie on two fixed straight lines $OP$, $OQ$ inclined to each other at an angle w. Prove that the locus of the circumcentre of $\mathrm{\Delta }\mathrm{O}\mathrm{A}\mathrm{B}$ is a circle. Find the locus of the lines which move parallel to the zx-plane and meet the curves:
$xy=c^2$, $z=0$;
$y^2=4cz$, $x=0$

[20M]

4) Two cones with a common vertex pass through the curves
$y=0,z^2=4ax$
$x=0,z^2=4by$

The plane $z=0$ meets them in two conics which intersect in four concyclic points. Show that vertex lies on the surface $z^2({\displaystyle \frac{x}{a}}+{\displaystyle \frac{y}{b}})=4(x^2+y^2)$.
Find the radii of curvature and torsion at any point of the curve $x^2+y^2=a^2$, $x^2-y^2=az$. (20)

[20M]