PYQs

1) Find the equations of the two generating lines through any point $(a\cos \theta ,b\sin \theta ,0)$ of the principal elliptic section ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}=1$, $z=0$ of the hyperboloid by the plane $z=0$.

[2014, 15M]

2) A variable generator meets two generators of the system through the extremities $B$ and $B^{\mathrm{\prime }}$ of the minor axis of the principal elliptic section of the hyperboloid ${\displaystyle \frac{x^2}{b^2}}-z^2{c}^2=1$ in $P$ and $P^{\mathrm{\prime }}$ prove that $BP\cdot P^{\mathrm{\prime }}{B}^{\mathrm{\prime }}=a^2+c^2$.

[2013, 20M]

3) Show that generators through any one of the ends of an equi-conjugate diameter of the principal elliptic section of the hyperboloid ${\displaystyle \frac{x^2}{a^2}}+{\displaystyle \frac{y^2}{b^2}}-{\displaystyle \frac{z^2}{c^2}}=1$ are inclined to each other at an angle of ${60}^{\circ }$ if $a^2+b^2=6c^2$. Find also the condition for the generators to be perpendicular to each other.

[2011, 20M]

4) Find the vertices of the skew quadrilateral formed by the four generators of the hyperboloid ${\displaystyle \frac{x^2}{4}}+y^2-z^2=49$ passing through $(10,5,1)$ and $(14,2,-2)$$.

[2010, 20M]

5) Show that the equation $x^2-5xy+y^2+8x-20y+15=0$ represents a hyperbola. Find the coordinates of its center and the length of its real semi-axes.

[2001, 12M]