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 \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1\), \(z=0\) 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 \(B^{\prime}\) of the minor axis of the principal elliptic section of the hyperboloid \(\dfrac{x^{2}}{b^{2}}-z^{2} c^{2}=1\) in \(P\) and \(P^{\prime}\) prove that \(B P \cdot P^{\prime} B^{\prime}=a^{2}+c^{2}\).

[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 \(\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}} - \dfrac{z^2}{c^2}=1\) are inclined to each other at an angle of \(60^{\circ}\) if \(a^{2}+b^{2}=6 c^{2}\). 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 \(\dfrac{x^{2}}{4}+y^{2}-z^{2}=49\) 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]