2004

1) Find the equation of the sphere for which the circle \(x^{2}+y^{2}+z^{2}+2 x-4 y+5=0\), \(x-2 y+3 z\) \(+1=0\) 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 \(3 x+2 y-z+2=0\) at the point \((1,-2,1)\) and cuts orthogonally the sphere \(x^{2}+y^{2}+z^{2}-4 x+6 y+4-0\).

[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 \(7 x+10 y-30=0,5 y-3 z=0\).

[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 \(\sqrt{f x}+\sqrt{8 y}+\sqrt{h z}=0\) represents a cone that fouches the co-ordinate planes and that the equation to its reciprocal cone is \(f y z+g z x+h x y=0\).

[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

\(a x^{2}+b y^{2}+2 z=0\) is
\(a b\left(x^{2}+y^{2}\right)+2(a+b) z=1\).

[10M]

5) Show that the enveloping cylinder of the ellipsoid \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{c^{2}}=1\) whose generators are parallel to the line \(\dfrac{x}{0}=\dfrac{y}{\pm \sqrt{a^{2}-b^{2}}}=\dfrac{z}{c}\) 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 \(\dfrac{x^{2}}{4}+\dfrac{y^{2}}{3}+\dfrac{z^{2}}{2}=1\) lie in the plane \(x+2 y+3 z=0,\) 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
\(\dfrac{b c}{\sqrt{b^{2}+c^{2}}} \cdot \dfrac{c a}{\sqrt{c^{2}+a^{2}}}\) and \(\dfrac{a b}{\sqrt{a^{2}+b^{2}}}\) where \(a\), \(b\), \(c\) are lengths of the edges of a parallelopiped.

[10M]

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

[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 \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\). Prove that the locus of the orthocentre of the triangle \(\mathrm{CPD}\) is the curve \(2\left(b^{2} y^{2}+a^{2} x^{2}\right)^{3}=\left(a^{2}-b^{2}\right)^{2}\left(b^{2} y^{2}-a^{2} x^{2}\right)\).

[13M]

3) If \(\dfrac{x}{1}=\dfrac{y}{2}=\dfrac{z}{3}\) represents one of the that mutually perpendicular generators of the cone \(5 y z-8 z x-3 x y=0\). Find the equations of the other two.

[13M]

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

[14M]

2000

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

[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 \(\mathrm{OABC}\) is \(\dfrac{p}{x}+\dfrac{q}{y}+\dfrac{r}{z}=1\).

[10M]

3) A straight line \(\mathrm{AB}\) 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 \(\Delta \mathrm{OAB}\) is a circle. Find the locus of the lines which move parallel to the zx-plane and meet the curves:
\(x y=c^{2}\), \(z=0\);
\(y^{2}=4 c z\), \(x=0\)

[20M]

4) Two cones with a common vertex pass through the curves
\(y=0, z^{2}=4 a x\)
\(x=0, z^{2}=4 b y\)

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

[20M]