2008

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

[10M]

2) Find the distance of the point (-2,3,-4) from the line ${\displaystyle \frac{x+2}{3}}={\displaystyle \frac{2y+3}{4}}={\displaystyle \frac{3z+4}{5}}$ measured parallel to the plane $4x+12y-3z+1=0$.

[10M]

3) Obtain the equation of the sphere that 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) Derive the equations to the planes that touch the surface $4x^2-5y^2+7z^2+13=0$ and are parallel to the plane $4x$ $+20y-21z=0$. Also determine the coordinates of the points of contact.

[10M]

5) Consider the section of the enveloping cone of the ellipsoid ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}+{\displaystyle \frac{z^2}{c^2}}=1$ by the plane $z=0.$ If the cone has its vertex at the point Pand if the section happens to be a rectangular hyperbola, show that the equation to the locus of the point $P$ is ${\displaystyle \frac{x^2+y^2}{a^2+b^2}}+{\displaystyle \frac{z^2}{c^2}}=1$.

[10M]

2007

1) If the three concurrent lines whose direction cosines are $(l_1,m_k,n_j),(l_2,m_2,n_2),(l_3,m_3,n_3)$ are coplanar, prove that
$\left[\begin{array}{lll}{1}_1& m_1& n_1\\ l_2& m_2& n_2\\ l_3& m_3& n_3\end{array}\right]=0$

[10M]

2) Find the equations of the three planes through the line

parallel to the axes.

[10M]

3) Prove that the shortest distance between the line

and any tangent to the ellipse $x^2{\sin }^2\theta +y^2=a^2$, $z=0$ is constant in length.

[10M]

4) The plane ${\displaystyle \frac{x}{a}}+{\displaystyle \frac{y}{b}}+{\displaystyle \frac{z}{c}}=1$ cut the axes in $A$, $B$, $C$. Find the equation of the cone whose vertex is origin and the guiding curve is the circle $ABC$.

[10M]

5) Find the equation of the cylinder generated ${\displaystyle \frac{x}{1}}={\displaystyle \frac{y}{2}}={\displaystyle \frac{z}{3}}$, the guiding curve being the conic $z=2$, $3x^2+4xy+5y^2=1$.

[10M]

2006

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

[10M]

2) Two straight lines ${\displaystyle \frac{x-{\alpha }_1}{l_1}}={\displaystyle \frac{y-{\beta }_1}{m_1}}={\displaystyle \frac{z-{\gamma }_1}{n_1}};{\displaystyle \frac{x-{\alpha }_2}{l_2}}={\displaystyle \frac{y-{\beta }_2}{m_2}}={\displaystyle \frac{z-{\gamma }_1}{n_2}}$ are cut by a third hine whose direction cosines are $\lambda$, $u$ and $v$. Show that the length $d$ intercepted on the third line is given by

$d\left|\begin{array}{ccc}{l}_1& m_1& n_1\\ l_2& m_2& n_2\\ \lambda & \mu & \nu \end{array}\right|$= $\left|\begin{array}{ccc}{\alpha }_1-{\alpha }_2& {\beta }_1-{\beta }_2& {\gamma }_1-\gamma 2\\ l_1& m_1& n_1\\ l_2& m_2& n_2\end{array}\right|$

Deduce the length of the shortest distance between the first two lines.

[16M]

3) Find the condition that the plane $lx+my+n=0$ be a tangent plane to the cone $a{x}^2+b{y}^2+$ $c{z}^2+2fyz+2gzx+2hxy=0$.

[12M]

4) Prove that the locus of the pole of the plane $lx+my+nz=p$ with respect to system of conicoids ${\displaystyle \frac{x^2}{a^2+k}}+{\displaystyle \frac{y^2}{b^2+k}}+{\displaystyle \frac{z^2}{c^2+k}}=1,$ where $k$ is a parameter, is a straight line perpendicular to the given plane.

[12M]

2005

1) Find the equations of the generators of the hyperboloid ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}-{\displaystyle \frac{z^2}{c^2}}=1$ through any point of the principal elliptic section ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^3}{b^2}}-{\displaystyle \frac{z^2}{c^2}}\mathrm{l}$; $z=0$.

[10M]

2) A variable plane is at a constant distance $\mathrm{p}$ from the origin and meets the axes in $\mathrm{A},\mathrm{B}$ and $\mathrm{C}$ 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]

3) Find the locus of the point of intersection of perpendicular generators of a hyperboloid of one sheet.

[10M]

4) Planes are drawn through a fixed point $(\alpha ,\beta ,\gamma )$ so that their sections of the paraboloid $a^2+b{y}^2=2z$ are rectangular hyperbolas. 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]

5) Show that the enveloping cylinder of the conicoid
$c{x}^2+b{y}^2+c{z}^7=1$ with generators perpendicular to $z$-axis meets the plane $z=0$ in parabolas.

[10M]