Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/234

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 * Transposing one term in each to the second member and dividing, we get

$\frac{x^2}{a^2}=\frac{y^2}{b^2}$.


 * Therefore, from (A), $$\frac{x^2}{a^2} = \frac{1}{2}$$ and $$\frac{y^2}{b^2} = \frac{1}{2}$$,


 * giving $$a = \pm x \sqrt{2}$$ and $$b = \pm y \sqrt{2}$$.


 * Substituting these values in (B), we get the envelope

$xy = \pm \frac{k}{2\pi}$,


 * a pair of conjugate rectangular hyperbolas (see last figure).

EXAMPLES 1. Find the envelope of the family of straight lines $$y = 2mx + m^4$$, $$m$$ being the variable parameter. Ans. $x = -2m^3$, $y = - 3m^4$; or $16y^3 +27x^4 = 0$.

2. Find the envelope of the family of parabolas $$y^2 = a(x - a)$$, $$a$$ being the variable parameter. Ans. $x = 2a$, $y = \pm a$; or $y = \pm {1 \over 2} x$.

3. Find the envelope of the family of circles $$x^2 + (y - \beta)^2 = r^2 $$, $$\beta$$ being the variable parameter. Ans. $x = \pm r$

4. Find the equation of the curve having as tangents the family of straight lines $$y = mx \pm \sqrt{a^2m^2 + b^2}$$, the slope $$m$$ being the variable parameter. Ans. The ellipse $b^2x^2 + a^2y^2 = a^2b^2$. 5. Find the envelope of the family of circles whose diameters are double ordinates of the parabola $$y^2 = 4px$$. Ans. The parabola $y^2 = 4p(p + x)$. 6. Find the envelope of the family of circles whose diameters are double ordinates of the ellipse $$b^2x^2 + a^2y^2 = a^2b^2$$. Ans. The ellipse $\frac{x^2}{a^2 + b^2} + \frac{y^2}{b^2} = 1$. 7. A circle moves with its center on the parabola $$y^2 = 4ax$$, and its circumference passes through the vertex of the parabola. Find the equation of the envelope of the circles. Ans. The [cissoid] $y^2(x + 2a) + x^3 = 0$. 8. Find the curve whose tangents are $$y = lx \sqrt{al^2 + bl + c}$$, the slope $$l$$ being supposed to vary. Ans. $4(ay^2 + bxy + cx^2 ) = 4ac - b^2$. 9 Find the evolute of the ellipse $$b^2x^2 + a^2y^2 = a^2b^2$$, taking the equation of normal in the form $$by = ax \tan \phi - (a^2 - b^2 ) \sin\phi$$, the eccentric angle $$\phi$$ being the parameter. Ans. $x = \frac{a^2-b^2}{a} \cos^3\phi$, $y = \frac{b^2-a^2}{a} \sin^3\phi$; or $(ax)^{2 \over 3} + (by)^{2 \over 3} = (a^2 - b^2)^{2 \over 3}$. 10. Find the evolute of the hypocycloid $$x^{2 \over 3} + y^{2 \over 3} = a^{2 \over 3}$$, the equation of whose normal is $$y\,\cos \tau - x\,\sin \tau = a\,\cos 2\tau$$, $$\tau$$ being the parameter. Ans. $(x + y)^{2 \over 3} + (x - y)^{2 \over 3} = 2a^{2 \over 3}$. -