2000

1) Find the equation of the sphere through the circle $x^2+y^2+$ $z^2=4,x+2y-z=2$ and the point (1,-1,1)

[10M]

2) A variable straight line always intersects the lines $x=c,y$ $=0;y=c,z=0;z=c,x=0$.Find the equations to its locus.

[10M]

3) Show that the locus of mid-points of chords of the cone $a{x}^2+b{y}^2+c{z}^2+2fyz+2gzx+2hxy=0$ drawn parallel to the line ${\displaystyle \frac{x}{l}}={\displaystyle \frac{y}{m}}={\displaystyle \frac{z}{n}}$ is the plane $(al+hm+gn)x+(hl+bm+fn)y+(gl+fm+cn)z=0$

[10M]

1999

1) If $P$ and $D$ are the ends of a pair of semiconjugate diameters of the ellipse ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$ show that the tangents at $P$ and $D$ meet on the ellipse ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=2$

[10M]

2) Find the equation of the cylinder whose generators touch the sphere $x^2+y^2+z^2=9$ and are perpendicular to the plane $x-y-3z=5.$

[10M]

3) Calculate the curvature and torsion at the point $u$ of the curve given by the parametric equations $x=a(3u-u^2)$ $y=3a{u}^2$ $z=a(3u+u^2)$

[10M]

1998

1) Find the locus of the pole of a chord of the conic ${\displaystyle \frac{1}{r}}=1+ecos\theta$ which subtends a constant angle $2\alpha$ at the focus.

[10M]

2) Show that the planes $ax+by+cz+d=0$ divides the join of $P_1\equiv (x_1,y_1,z_1),P_2\equiv (x_2,y_2,z_2)$ in the ratio -${\displaystyle \frac{a{x}_1+b{y}_1+c{z}_1+d}{a{x}_2+b{y}_2+c{z}_2+d}}$. Hence show the planes $Uax+by+cz+d=0=a^{\prime }x+b^{\prime }y+c^{\prime }z+d^{\prime }=v$, $u+\lambda v=0$ and $u-\lambda v=0$ divide any transversal harmonically.

[10M]

3) Prove that a curve $x(s)$ is a generalized helix if and only if it satisfies the identity $x^{″}\cdot x^{‴}\times x^{iv}=0$.

TBC

[10M]

4) Find the smallest sphere (i.e., the sphere of smallest radius ) which touches the lines ${\displaystyle \frac{x-5}{2}}={\displaystyle \frac{y-2}{-1}}={\displaystyle \frac{z-5}{-1}}$ and ${\displaystyle \frac{x+4}{-3}}={\displaystyle \frac{y+5}{-6}}={\displaystyle \frac{z-4}{4}}$.

[10M]

5) Find the co-ordinates of the point of intersection of the generators ${\displaystyle \frac{x}{a}}-{\displaystyle \frac{y}{b}}-2\lambda =0={\displaystyle \frac{x}{a}}+{\displaystyle \frac{y}{b}}-{\displaystyle \frac{z}{\lambda }}$\ ${\displaystyle \frac{x}{a}}+{\displaystyle \frac{y}{b}}-2\mu =0={\displaystyle \frac{x}{a}}-{\displaystyle \frac{y}{b}}-{\displaystyle \frac{z}{\mu }}$ of the surface ${\displaystyle \frac{x^2}{a^2}}-{\displaystyle \frac{y^2}{b^2}}=2z$. Hence show that thelocus of the points of intersection of perpendicular generators is the curve of intersection of the surface with plane $2z+(a^2-b^2)=0$.

[10M]

6) Let $P\equiv (x^{\prime },y^{\prime },z^{\prime })$ lie on the ellipsoid ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}+{\displaystyle \frac{z^2}{c^2}}=1$ If the length of the normal chord through $P$ is equal to $4PG$ where $G$ is the intersection of the normal with the z-plane, then show that $P$ lies on the cone ${\displaystyle \frac{x^2}{a^6}}(2c^2-a^2)+{\displaystyle \frac{y^2}{b^6}}(2c^2-b^6)+{\displaystyle \frac{z^2}{c^4}}=0$\

[10M]

1997

1) Let $\mathrm{P}$ be a point on an ellipse with its centre at the point $\mathrm{C}$. Let $\mathrm{C}\mathrm{D}$ and $\mathrm{C}\mathrm{P}$ be two conjugate diameters If the normal at $\mathrm{P}$ cuts $\mathrm{C}\mathrm{D}$ in $\mathrm{F},$ show that $\mathrm{C}\mathrm{D}\cdot \mathrm{P}\mathrm{F}$ is a constant and the locus of $\mathrm{F}$ is ${\displaystyle \frac{a^2}{x^2}}+{\displaystyle \frac{b^2}{y^2}}={\left[{\displaystyle \frac{a^2-b^2}{x^2+y^2}}\right]}^2$ where ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$ equation of the given ellipse.

[10M]

2) A circle passing through the focus of conic section whose latus rectum is $2l$ meets the conic in four points whose distances from the focus are ${\gamma }_1,{\gamma }_2,{\gamma }_3$ and ${\gamma }_4$ Prove that ${\displaystyle \frac{1}{{\gamma }_1}}+{\displaystyle \frac{1}{{\gamma }_2}}+{\displaystyle \frac{1}{{\gamma }_3}}+{\displaystyle \frac{1}{{\gamma }_4}}={\displaystyle \frac{2}{l}}$

[10M]

3) Determine the curvature of the circular helix $y(t)=(a\cos t)\hat{i}+(a\sin t)\hat{j}+(bt)\hat{k}$ and an equation of the normal plane at the point $(0,a,{\displaystyle \frac{\pi b}{2}})$.

[10M]

4) Find the reflection of the plane $x+y+2-1-0$ in plane $3x+4z+1=0$.

[10M]

5) Show that the point of intersection of three mutually perpendicular tangent planes to the ellipsoid ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}+{\displaystyle \frac{z^2}{c^2}}=1$ lies on the sphere $x^2+y^2+z^2=a^2+b^2+c^2$.

[10M]

6) Find the equation of the spheres which pass through the circle $x^2+y^2+z^2-4x-y+3z+12=0$,$2x+3y-7z=10$ and touch the plane $x-2y+2z=1$.

[10M]

1996

1) Find the equation of the common tangent to the parabolas $y^2=4ax$ and $x^2=4by$.

[15M]

2) If the normal at any point ${}^{\mathrm{\prime }}{\mathrm{t}}_1{}^{\mathrm{\prime }}$ of a. rectangular hyperbola $\mathrm{x}\mathrm{y}={\mathrm{c}}^2$ meets the curve again at the point ${}^{\prime }{t}_2^{\prime }$ prove that ${\mathrm{t}}_1^3{\mathrm{t}}_2=-1$.

[15M]

3) A variable plane is at a constant distance $p$ from the origin and meets the axes in $A,B$ and $C$. Through $A,B,C$ the planes are drawn parallel to the coordinate planes. Show that the locus of their point of intersection is given by ${\mathrm{x}}^{-2}+{\mathrm{y}}^{-2}+{\mathrm{z}}^{-2}={\mathrm{p}}^{-2}$

[15M]

4) Find the equation of the sphere which passes through the points (1,0,0),(0,1,0),(0,0,1) and has the smallest possible radius.

[15M]

5) The generators through a point $\mathrm{P}$ on the hyperboloid ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}-{\displaystyle \frac{z^2}{c^2}}=1$ meet the principal elliptic section in two points such that the eccentric angle of one is double that of the other. Show that P lies on the curve $x={\displaystyle \frac{a(1-3t^2)}{1+t^2}},y={\displaystyle \frac{bt(3-t^2)}{1+t^2}},z=ct$

[15M]

6) A curve is drawn on a right circular cone, semi-vertical angle $\alpha ,$ so as to cut all the generators at the same angle $\beta$. Show that its projection on a plane at right angles to the axis is an equiangular spiral, Find expressions for its curvature and torsion.

[15M]

1995

1) Two conjugate semi-diameters of the ellipse ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$ cut the circle $x^2+y^2-r^2$ at $P$ and $Q$. Show that the locus of the middle point of $\mathrm{P}\mathrm{Q}$ is $a^2\{{(x^2+y^2)}^2-r^2{x}^2\}+b^2\{{(x^2+y^2)}^2-r^2{y}^2\}=0$

[20M]

2) If the normal at one of the extremities of latus rectum of the conic ${\displaystyle \frac{1}{r}}=1+e\cos \theta ,$ meets the curve again at Q. show that $SQ={\displaystyle \frac{l(1+3e^2+e^4)}{(1+e^2-e^4)}}$ where $\mathrm{S}$ is the focus of the conic

[20M]

3) Through a point $\mathrm{P}({\mathrm{x}}^{\mathrm{\prime }},{\mathrm{y}}^{\mathrm{\prime }},z^2)\mathrm{a}$ plane is drawn at right angles to OP to meet the coordinate axes in $\mathrm{A},\mathrm{B},\mathrm{C}$. Prove that the area of the triangle $\mathrm{A}\mathrm{B}\mathrm{C}$ is ${\displaystyle \frac{r^5}{2x^{\mathrm{\prime }}{y}^{\mathrm{\prime }}{z}^{\mathrm{\prime }}}},$ where $\mathrm{r}$ is the measure of OP.

[20M]

4) Two spheres of radii $r_1$ and $r_2$ cut orthogonally, Prove that the area of the common circle is ${\displaystyle \frac{\pi r_1^2{r}_2^2}{r_1^2+r_2^2}}$

[20M]

5) Show that a plane through one member of the $\lambda$ -system and one member of $\mu$ -system is tangent plane to the hyperboloid at the point of intersection of the two generators.

[20M]

6) Prove that the parallels through the origin to the binormals of the helix $x=a\cos \theta ,y=\mathrm{asin}\theta ,z=k\theta$ lie upon the right cone $a^2(x^2+y^2)=k^2{z}^2$

[20M]