2000

1) Find the equation of the sphere through the circle \(x^{2}+y^{2}+\) \(z^{2}=4, x+2 y-z=2\) 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 \(a x^{2}+b y^{2}+c z^{2}+2 f y z+2 g z x+2 h x y=0\) drawn parallel to the line \(\dfrac{x}{l}=\dfrac{y}{m}=\dfrac{z}{n}\) is the plane \((a l+h m+g n) x+(h l+b m+f n) y+(g l+f m+c n) z=0\)

[10M]

1999

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

[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 \(x=a(3u-u^2)\) \(y=3au^2\) \(z=a(3u+u^2)\)

[10M]

1998

1) Find the locus of the pole of a chord of the conic \(\dfrac{1}{r}=1+e cos\theta\) 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 -\(\dfrac{ax_1+by_1+cz_1+d}{ax_2+by_2+cz_2+d}\). Hence show the planes \(U ax+by+cz+d=0=a'x+b'y+c'z+d'= v\), \(u+\lambda v=0\) 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 \(\dfrac{x-5}{2}=\dfrac{y-2}{-1}=\dfrac{z-5}{-1}\) and \(\dfrac{x+4}{-3}=\dfrac{y+5}{-6}=\dfrac{z-4}{4}\).

[10M]

5) Find the co-ordinates of the point of intersection of the generators \(\dfrac{x}{a}-\dfrac{y}{b}-2\lambda=0=\dfrac{x}{a}+\dfrac{y}{b}-\dfrac{z}{\lambda}\)\ \(\dfrac{x}{a}+\dfrac{y}{b}-2\mu=0=\dfrac{x}{a}-\dfrac{y}{b}-\dfrac{z}{\mu}\) of the surface \(\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=2z\). 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 \(P\equiv(x’,y’,z’)\) lie on the ellipsoid \(\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=1\) 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 \(\dfrac{x^2}{a^6}(2c^2-a^2)+\dfrac{y^2}{b^6}(2c^2-b^6)+\dfrac{z^2}{c^4}=0\)\

[10M]

1997

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

[10M]

2) A circle passing through the focus of conic section whose latus rectum is \(2 l\) meets the conic in four points whose distances from the focus are \(\gamma_{1}, \gamma_{2}, \gamma_{3}\) and \(\gamma_{4}\) Prove that \(\dfrac{1}{\gamma_{1}}+\dfrac{1}{\gamma_{2}}+\dfrac{1}{\gamma_{3}}+\dfrac{1}{\gamma_{4}}=\dfrac{2}{l}\)

[10M]

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

[10M]

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

[10M]

5) Show that the point of intersection of three mutually perpendicular tangent planes to the ellipsoid \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{c^{2}}=1\) 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 \(x^{2}+y^{2}+z^{2}-4 x-y+3 z+12=0\),\(2 x+3 y-7 z=10\) and touch the plane \(x-2 y+2 z=1\).

[10M]

1996

1) Find the equation of the common tangent to the parabolas \(y^{2}=4 a x\) and \(x^{2}=4 b y\).

[15M]

2) If the normal at any point \({ }^{\prime} \mathrm{t}_{1}{ }^{\prime}\) of a. rectangular hyperbola \(\mathrm{xy}=\mathrm{c}^{2}\) meets the curve again at the point \('t_{2}'\) 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 \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}-\dfrac{z^{2}}{c^{2}}=1\) 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 \(x=\dfrac{a\left(1-3 t^{2}\right)}{1+t^{2}}, y=\dfrac{b t\left(3-t^{2}\right)}{1+t^{2}}, z=c t\)

[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 \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\) cut the circle \(x^{2}+y^{2}-r^{2}\) at \(P\) and \(Q\). Show that the locus of the middle point of \(\mathrm{PQ}\) is \(a^{2}\left\{\left(x^{2}+y^{2}\right)^{2}-r^{2} x^{2}\right\}+b^{2}\left\{\left(x^{2}+y^{2}\right)^{2}-r^{2} y^{2}\right\}=0\)

[20M]

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

[20M]

3) Through a point \(\mathrm{P}\left(\mathrm{x}^{\prime}, \mathrm{y}^{\prime}, z^{2}\right) \mathrm{a}\) 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 \(\mathrm{ABC}\) is \(\dfrac{r^{5}}{2 x^{\prime} y^{\prime} z^{\prime}},\) 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 \(\dfrac{\pi r_{1}^{2} r_{2}^{2}}{r_{1}^{2}+r_{2}^{2}}\)

[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 \(x=a \cos \theta, y=\operatorname{asin} \theta, z=k \theta\) lie upon the right cone \(a^{2}\left(x^{2}+y^{2}\right)=k^{2} z^{2}\)

[20M]