91Ó°ÊÓ

Use a suitable rotation of axes to simplify the equation of the curve $$ 4 x^{2}+24 x y+11 y^{2}=5 $$

A circle of radius \(b\) rolls without sliding on the outside of a fixed circle of radius nb. The locus of a point fixed on the circumference of the rolling circle is called an epicycloid (when \(n\) is an integer, an \(n\)-cusped epicycloid). Obtain the parametric equations

If \((X, Y, Z)\) is outside the sphere \(x^{2}+y^{2}+z^{2}=k^{2}\), its polar plane contains the points of contact of all the tangent planes that pass through \((X, Y, Z)\).

What happens to the equation for the rectangular hyperbola \(x^{2}-y^{2}=a^{2}\) when we rotate the axes through the angle \(\frac{1}{4} \pi\) ?

A circle of radius \(b\) rolls without sliding on the inside of a fixed circle of radius \(n b\), where \(n>1\). Find parametric equations for the hypocycloid (when \(n\) is integral, the \(n\)-cusped hypocycloid) which is the locus of a point fixed on the circumference of the rolling circle. Sketch the cases \(n=2\) (which is surprising), \(n=3\) (the deltoid), and \(n=4\) (the astroid). Fliminate the parameter in the last two cases, obtaining, for the astroid, $$ x^{2 / 3}+y^{2 / 3}=a^{2 / 3} \quad(a=4 b) $$

The circle \(r=l\) inverts the conic \(8.42\) into the limaçon $$ r=l(1+\varepsilon \cos \theta) \text {. } $$ Sketch this curve for various values of \(\varepsilon\). When \(\varepsilon=1\) (so that the conic is a parabola), it is a cardioid.