1994

1) If $2\varphi$ be the angle between the tangents from $P(x_1,y_1)$ to ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1,$ prove that ${\lambda }_1{\cos }^2\varphi +{\lambda }_2{\sin }^2\varphi =0$ where ${\lambda }_1,{\lambda }_2,$ are the parameters of two confocals to the ellipse through $P$

[10M]

2) If the normals at the points $\alpha ,\beta ,\gamma ,\delta$ on the conic $1/r=1+e\cos \theta$ meet at $(\rho ,\varphi ),$ prove that $\alpha +\beta +\gamma +\delta -2\varphi =$ odd multiple of $\pi$ radians.

[10M]

3) A variable plane is at a constant distance $p$ from the origin $O$ and meets the axes in $A,B$ and C. Show that the locus of the centroid of the tetrahedron $OABC$ is ${\displaystyle \frac{1}{x^2}}+{\displaystyle \frac{1}{y^2}}+{\displaystyle \frac{1}{z^2}}={\displaystyle \frac{16}{p^2}}$

[10M]

4) Find the equations to the generators of hyperboloid, through any point of the principal elliptic section ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}-{\displaystyle \frac{z^2}{c^2}}=1,z=0$

[10M]

5) Planes are drawn through a fixed point $(\alpha ,\beta ,\gamma )$ so that their sections of the paraboloid $a{x}^2+b{y}^2=2z$ are rectangular hy perbolas. Prove that they touch the cone. ${\displaystyle \frac{(x-{\alpha }^2)}{b}}+{\displaystyle \frac{(y-\beta {)}^2}{a}}+{\displaystyle \frac{(z-\gamma {)}^2}{a+b}}=0$

[10M]

6) Find $f(\theta )$ so that the curve $x=a\cos \theta ,y=a\sin \theta ,z=f(\theta )$ determines a plane curve.

[10M]

1993

1) If $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ represents a pair of lines, prove that the area of the triangle formed by their bisectors and axis of $x$ is $\sqrt{{\displaystyle \frac{(a-b{)}^2+4h^2}{2h}}},{\displaystyle \frac{ca-g^2}{ab-h^2}}$

[10M]

2) Find the equation of the director circle of the conic $l/r=1+e\cos \theta$ and also obtain the asymptotes of the above conic.

[10M]

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

[10M]

4) Prove that the centres of the spheres which touch the lines $y=mx,z=c;y=-mx,z=-c$ lie upon the conicoid mxy $+cr(1+m^2)=0$.

[10M]

5) Find the locus of the point of intersection of perpendicular generator of a hyperboloid of one sheet.

[10M]

6) A curve is drawn on a parabolic cylinder so as to cut all the generators at the same angle. Find its curvature and torsion.

[10M]

1992

1) If $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ represents two intersecting straight lines, show that the square of the distance of the point of intersection of the straight lines from the origin is

[10M]

2) Discuss the nature of the conic

in details.

[10M]

3) A straight line, always parallel to the plane of $yz$ passed through the curves $x^2+y^2=a^2,z=0$ and $x^2=ax$, $y=0$; prove that the equation of the surface generated is

[10M]

4) Tangent planes are drawn to the ellipsoid
${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}+{\displaystyle \frac{z^2}{c^2}}=1$

through the point $(\alpha ,\beta ,\gamma )$. Prove that the perpendiculars to them from the origin generate the cone

[10M]

5) Show that the locus of the foot of the perpendicular from the centre to the plane through the extremities of three conjugate semi-diameters of the ellipsoid ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}+{\displaystyle \frac{z^2}{c^2}}=1$ is

[10M]

6) Define an oscillating plane and derive its equation in vector form. If the tangent and binormal at a point $\mathrm{P}$ of the curve make angles $\theta ,\varphi$ respectively with the fixed direction, show that

where $\mathrm{k}$ and $\tau$ are respectively curvature and torsion of the curve at $\mathrm{P}$.

[10M]

1991

1) Prove that the locus of a line which meets the lines $y=mx$, $z=c$ and $y=-mx$, $z=-c$ and also meets the circle $x^2+y^2=a^2$, $z=0$ is

2) Four generators of the hyperboloid ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}+{\displaystyle \frac{z^2}{c^2}}=1$ form a skew quadrilateral whose vertices are the points $(a\cos {\theta }_i\sec {\varphi }_i,b\sin {\theta }_i\sec {\varphi }_i,c\tan {\varphi }_i)$; $i=1,2,3,4$.

Prove that ${\theta }_1+{\theta }_3={\theta }_2+{\theta }_4,{\varphi }_1+{\varphi }_3={\varphi }_2+{\varphi }_4$

1990

1.(a) What is meant by the direction cosines of a line in 3 - space? Show that the equation of any line can be written in the form ${\displaystyle \frac{x-\alpha }{l}}={\displaystyle \frac{y-\beta }{m}}={\displaystyle \frac{z-\gamma }{n}}$ explaining the meaning of the parameters involved.

1.(b) If ${\displaystyle \frac{x-{\alpha }^{\mathrm{\prime }}}{l^{\mathrm{\prime }}}}={\displaystyle \frac{y-{\beta }^{\mathrm{\prime }}}{m^{\mathrm{\prime }}}}={\displaystyle \frac{z-{\gamma }^{\mathrm{\prime }}}{n^{\mathrm{\prime }}}}$ is another line which is skew to (that in the above problem (a)), find the length of their common perpendicular, without knowing that the two lines are skew, how will you determine whether they are coplanar?

1989

1) Prove that the planes $ny-mz=\lambda$, $lz-nx=\mu$, $mx-ly=\gamma$ have a common line if $1\lambda +m\mu +n\gamma =0$. Show also that the distance of the line from the origin is ${\left({\displaystyle \frac{{\lambda }^2+{\mu }^2+{\gamma }^2}{l^2+m^2+n^2}}\right)}^{1/2}$.

2) Show that if all the plane sections of a surface which has equation of second degree, are circles, the surface must be a sphere.

3) Show that the locus of the point of intersection of three mutually perpendicular tangent planes to the paraboloid $a{x}^2+b{y}^2=2z$ is a plane perpendicular to the axis of the paraboloid.

4) Find the locus of the perpendiculars from the origin on the tangent planes to the ellipsoid ${\displaystyle \frac{-x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}+{\displaystyle \frac{z^2}{c^2}}=1,$ which cut off from the axes intercepts, the sum of whose reciprocals is equal to a constant $1/\mathrm{K}$.