Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/235

11. Find the envelope of the circles which pass through the origin and have their centers on the hyperbola $x^{2}-y^{2}=c^{2}$.

Ans. The lemniscate $(x^{2}+y^{2})^{2}=a^{2}(x^{2}-y^{2})$.

12. Find the envelope of a line such that the sum of its intercepts on the axes equals $c$.

Ans. The parabola $x^{1 \over 2}+y^{1 \over 2}=c^{1 \over 2}$.

13. Find the equation of the envelope of the system of circles $x^{2}+y^{2}-2(a+2)x+a^{2}=0$, where $a$ is the parameter. Draw a figure illustrating the problem.

Ans. $y^{2}=4x$.

14. Find the envelope of the family of ellipses $b^{2}x^{2}+a^{2}y^{2}=a^{2}b^{2}$, when the sum of its semiaxes equals $c$.

Ans. The hypocycloid $x^{2 \over 3}+y^{2 \over 3}=c^{2 \over 3}$.

15. Find the envelope of the ellipses whose axes coincide, and such that the distance between the extremities of the major and minor axes is constant and equal to $l$.

Ans. A square whose sides are $(x\pm y)^{2}=l^{2}$.

16. Projectiles are fired from a gun with an initial velocity $v_{o}$. Supposing the gun can be given any elevation and is kept always in the same vertical plane, what is the envelope of all possible trajectories, the resistance of the air being neglected?

Hint. The equation of any trajectory is

$y=x\,\tan \alpha -{\frac {gx^{2}}{2v_{o}^{2}\cos ^{2}\alpha }}$,

$\alpha$ being the variable parameter.

Ans. The parabola $y={\frac {v_{o}^{2}}{2g}}-{\frac {gx^{2}}{2v_{o}^{2}}}$.

17. Find the equation of the envelope of each of the following family of curves, $t$ being the parameter; draw the family and the envelope :

(a) $(x-t)^{2}+y^{2}=1-t^{2}$.	(i) $(x-t)^{2}+y^{2}=4t$.
(b) $x^{2}+(y-t)^{2}=2t$.	(j) $x^{2}+(y-t)^{2}=4-t^{2}$ .
(c) $(x-t)^{2}+y^{2}={1 \over 2}t^{2}-1$.	(k) $(x-t)^{2}+(y-t)^{2}=t^{2}$.
(d) $x^{2}+(y-t)^{2}={1 \over 4}t^{2}$.	(l) $(x-t)^{2}+(y+t)^{2}=t^{2}$.
(e) $y=tx+t^{2}$.	(m) $y=t^{2}x+t$.
(f) $x=2ty+t^{4}$.	(n) $y=t(x-2t)$.
(g) $y=tx+{1 \over t}$.	(o) $x={y \over t}+t$.
(h) $y^{2}=t(x+2t)$.	(p) $(x-t)^{2}+4y^{2}=t$.