IFoS PYQs 4
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 \(\dfrac{x+2}{3}=\dfrac{2 y+3}{4}=\dfrac{3 z+4}{5}\) measured parallel to the plane \(4 x+12 y-3 z+1=0\).
[10M]
3) Obtain the equation of the sphere that touches the plane \(3 x+2 y-z+2=0\) at the point \((1,-2,1)\) and cuts orthogonally the sphere \(x^{2}+y^{2}+z^{2}-4 x+6 y+4=0\).
[10M]
4) Derive the equations to the planes that touch the surface \(4 x^{2}-5 y^{2}+7 z^{2}+13=0\) and are parallel to the plane \(4 x\) \(+20 y-21 z=0\). Also determine the coordinates of the points of contact.
[10M]
5) Consider the section of the enveloping cone of the ellipsoid \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}+\dfrac{z^{2}}{c^{2}}=1\) 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 \(\dfrac{x^{2}+y^{2}}{a^{2}+b^{2}}+\dfrac{z^{2}}{c^{2}}=1\).
[10M]
2007
1) If the three concurrent lines whose direction cosines are \(\left(l_{1}, m_{k}, n_{j}\right),\left(l_{2}, m_{2}, n_{2}\right),\left(l_{3}, m_{3}, n_{3}\right)\) 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
\[\dfrac{x-1}{2}=\dfrac{y-2}{3}=\dfrac{z-3}{4}\]parallel to the axes.
[10M]
3) Prove that the shortest distance between the line
\[z=x \tan \theta, y=0\]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 \(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=1\) 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 \(\dfrac{x}{1}=\dfrac{y}{2}=\dfrac{z}{3}\), the guiding curve being the conic \(z=2\), \(3 x^{2}+4 x y+5 y^{2}=1\).
[10M]
2006
1) Prove that the locus of a line which meets the two lines \(y=\pm m x, z=\pm c\) and the circle \(x^{2}+y^{2}=a^2\), \(z=0\) is
\[\mathrm{c}^{2} \mathrm{m}^{2}(\mathrm{cy} -\mathrm{mzx})^{2}+\mathrm{c}^{2}(\mathrm{yz}-\mathrm{cmx})^{2}=\mathrm{a}^{2} \mathrm{m}^{2}\left(\mathrm{z}^{2}-\mathrm{c}^{2}\right)^{2}\][10M]
2) Two straight lines \(\dfrac{x-\alpha_{1}}{l_{1}}=\dfrac{y-\beta_{1}}{m_{1}}=\dfrac{z-\gamma_{1}}{n_{1}} ; \dfrac{x-\alpha_{2}}{l_{2}}=\dfrac{y-\beta_{2}}{m_{2}}=\dfrac{z-\gamma_{1}}{n_{2}}\) 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
\(d \begin{vmatrix} l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \\ \lambda & \mu & \nu \end{vmatrix}\)= \(\begin{vmatrix} \alpha_1-\alpha_2 & \beta_1-\beta_2 & \gamma_1 - \gamma2 \\ l_1 & m_1 & n_1 \\ l_2 & m_2 & n_2 \end{vmatrix}\)
Deduce the length of the shortest distance between the first two lines.
[16M]
3) Find the condition that the plane \(l x+m y+n=0\) be a tangent plane to the cone \(a x^{2}+b y^{2}+\) \(c z^{2}+2 f y z+2 g z x+2 h x y=0\).
[12M]
4) Prove that the locus of the pole of the plane \(l x+m y + n z=p\) with respect to system of conicoids \(\dfrac{x^{2}}{a^{2}+k}+\dfrac{y^{2}}{b^{2}+k}+\dfrac{z^{2}}{c^{2}+k}=1,\) 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 \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}-\dfrac{z^{2}}{c^{2}}=1\) through any point of the principal elliptic section \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{3}}{b^{2}}-\dfrac{z^{2}}{c^{2}} \mathrm{l}\); \(z=0\).
[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 \(\dfrac{1}{x^{2}}+\dfrac{1}{y^{2}}+\dfrac{1}{z^{2}}=\dfrac{16}{p^{2}}\).
[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 \(a^{2}+b y^{2}=2 z\) are rectangular hyperbolas. Prove that they touch the cone \(\dfrac{(x-\alpha)^{2}}{b}+\dfrac{(y-\beta)^{2}}{a}+\dfrac{(z-\gamma)^{2}}{a+b}=0\).
[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]