1988

1) A sphere of constant radius \(\mathrm{K}\) passes through the origin and meets the axes in \(\mathrm{A}, \mathrm{B}, \mathrm{C}\). prove that the centroid of the triangle \(\mathrm{ABC}\) lies on the sphere \(9\left(x^{2}+y^{2}+z^{2}\right)=4 K^{2}\).

2) Prove that, in general, from a point \((\alpha, \beta, \gamma)\) five normals can be drawn to the paraboloid \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=2 z\) and that these normals lie on the cone \(\dfrac{\alpha}{x-\alpha}-\dfrac{\beta}{y-\beta}+\dfrac{a^{2}-b^{2}}{z-\gamma}=0\).

3) Show that the locus of the equal conjugate semidiameters of the ellipsoid \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{c^{2}}=1\) is \(\dfrac{\left(b^{2}+c^{2}-2 a^{2}\right)}{a^{2}} x^{2}\)+\(\dfrac{\left(c^{2}+a^{2}-2 b^{2}\right)}{b^{2}} y^{2}\)+\(\dfrac{\left(a^{2}+b^{2}-2 c^{2}\right)}{c^{2}} z^{2}=0\)

4) Find the locus of the points of intersection of perpendicular generators of the hyperboloid \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}-\dfrac{z^{2}}{c^{2}}=1\).

1987

1) Find the locus of a point which is equidistant from the lines \(y=mx\), \(z=c\); \(y=-m x\), \(z=-c\).

2) The direction cosines \((1, m, n)\) of a line are connected by the relations \(2 l+2 m+n=0\), \(3 l^{2}+5 m^{2}=5 n^{2}\). Show that two such lines are possible and that they are perpendicular to each other.

3) A sphere through the origin cuts the co - ordinate axes in the points \(\mathrm{A}, \mathrm{B}, \mathrm{C}\) and the plane through \(\mathrm{A}\), \(\mathrm{B}\), \(\mathrm{C}\) contains the point \((1,-2,-3)\). Show that the locus of the centre of the sphere is \(\dfrac{1}{x}-\dfrac{2}{y}-\dfrac{3}{z}=2\)

1986

1) Find the equation of the plane passing through the line \(\dfrac{x}{2}=\dfrac{y}{3}=\dfrac{z}{-1}\) and perpendicular to the plane containing the lines \(\dfrac{x}{3}=\dfrac{y}{-1}=\dfrac{z}{2}\) and \(\dfrac{x}{-1}=\dfrac{y}{2}=\dfrac{z}{3}\).

2) Find an expression for the product of distances from the origin to the points of intersection of the quadric \(ax^{2}\)+\(by^{2}\)+\(cz^{2}\)+\(2fyz\)+\(2gzx\)+\(2hxy\)+\(2ux\)+\(2vy\)+\(2wz\)+\(d=0\) with the line through the origin with direction cosines \(l\), \(m\), \(n\).

3) Any three mutually orthogonal lines drawn through a point \(\mathrm{C}(0,1,-1)\) meet the quadric \(2 x^{2}+3 y^{2}+5 z^{2}=1\) in points

1985

1) Prove that the perpendicular distance of the point \(\left(x_{1}, y_{1}, z_{1}\right)\) from the plane \(a x+b y+c z+d=0\) is \(\dfrac{a x_{1}+b y_{1}+c z_{1}+d}{\sqrt{a^{2}+b^{2}+c^{2}}}\).

2) A plane passes through a fixed point \((a, b, c)\) and cuts the axes in \(\mathrm{A}, \mathrm{B}, \mathrm{C}\). show that the locus of the centre of the sphere OABC, where \(\mathrm{O}\) is the origin is \(\dfrac{a}{x}+\dfrac{b}{y}+\dfrac{c}{z}=2\).

3) Find the equation of the cone whose vertex \((1,1,1)\) and the base circle \(x^{2}+y^{2}=4\), \(z=2\).

1984

1) Find the condition that the plane \(l x+m y+n z=p\) should touch the conicoid \(a x^{2}+b y^{2}+c z^{2}=1\). Hence find locus of the point of intersection of three mutually perpendicular tangent planes to a central conicoid.

2) Show that the sum of the squares of the reciprocals of any three diameters of an ellipsoid which are mutually at right angles is constant.

3) Show that the locus of the line of intersection of tangent planes to the cone \(a x^{2}+b y^{2}+c z^{2}=0\) which touch along perpendicular generators is the cone \(a^{2}(b+c) x^{2}+b^{2}(c+a) y^{2}+c^{2}(a+b) z^{2}=0\).

1983

1) Find the locus of the line which intersects the three lines \(y-z=1\), \(x=0\); \(z-x=1\), \(y=0\); \(x-y=1\); \(z=0\).

2) Prove that the locus of points from which three mutually perpendicular planes can be drawn to touch the ellipse \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1, z=0\) is the sphere

3) Prove that the feet of the six normals drawn to the ellipsoid \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{c^{2}}=1\) from any point \((\alpha, \beta, \gamma)\) be on the curve of intersection of the ellipsoid and the cone

\(\dfrac{a^{2}\left(b^{2}-c^{2}\right) \alpha}{x}\)+\(\dfrac{b^{2}\left(c^{2}-a^{2}\right) \beta}{y}\)+\(\dfrac{c^{2}\left(a^{2}-b^{2}\right) \gamma}{z}=0\)