Chapter 11

The width of a parabola at the focus Show that the number 4\(p\) is the width of the parabola \(x^{2}=4 p y(p>0)\) at the focus by showing that the line \(y=p\) cuts the parabola at points that are 4\(p\) units apart.

Graph the lines and conic sections in Exercises \(65-74.\) $$r=1 /(1+\cos \theta)$$

The asymptotes of \(\left(x^{2} / a^{2}\right)-\left(y^{2} / b^{2}\right)=1\) Show that the vertical distance between the line \(y=(b / a) x\) and the upper half of the right-hand branch \(y=(b / a) \sqrt{x^{2}-a^{2}}\) of the hyperbola \(\left(x^{2} / a^{2}\right)-\left(y^{2} / b^{2}\right)=1\) approaches 0 by showing that $$ \lim _{x \rightarrow \infty}\left(\frac{b}{a} x-\frac{b}{a} \sqrt{x^{2}-a^{2}}\right)=\frac{b}{a} \lim _{x \rightarrow \infty}\left(x-\sqrt{x^{2}-a^{2}}\right)=0 $$ Similar results hold for the remaining portions of the hyperbola and the lines \(y=\pm(b / a) x .\)

Area Find the dimensions of the rectangle of largest area that can be inscribed in the ellipse \(x^{2}+4 y^{2}=4\) with its sides parallel to the coordinate axes. What is the area of the rectangle?

Volume Find the volume of the solid generated by revolving the region enclosed by the ellipse \(9 x^{2}+4 y^{2}=36\) about the (a) \(x\) -axis, (b) \(y\) -axis.

Graph the lines and conic sections in Exercises \(65-74.\) $$r=1 /(1+2 \cos \theta)$$

Volume The "triangular" region in the first quadrant bounded by the \(x\) -axis, the line \(x=4,\) and the hyperbola \(9 x^{2}-4 y^{2}=36\) is revolved about the \(x\) -axis to generate a solid. Find the volume of the solid.

Tangents Show that the tangents to the curve \(y^{2}=4 p x\) from any point on the line \(x=-p\) are perpendicular.

Tangents Find equations for the tangents to the circle \((x-2)^{2}+(y-1)^{2}=5\) at the points where the circle crosses the coordinate axes.

Volume The region bounded on the left by the \(y\) -axis, on the right by the hyperbola \(x^{2}-y^{2}=1,\) and above and below by the lines \(y=\pm 3\) is revolved about the \(y\) -axis to generate a solid. Find the volume of the solid.

Centroid Find the centroid of the region that is bounded below by the \(x\) -axis and above by the ellipse \(\left(x^{2} / 9\right)+\left(y^{2} / 16\right)=1\)

Surface area The curve \(y=\sqrt{x^{2}+1}, 0 \leq x \leq \sqrt{2},\) which is part of the upper branch of the hyperbola \(y^{2}-x^{2}=1,\) is revolved about the \(x\) -axis to generate a surface. Find the area of the surface.

The reflective property of parabolas The accompanying figure shows a typical point \(P\left(x_{0}, y_{0}\right)\) on the parabola \(y^{2}=4 p x\) . The line \(L\) is tangent to the parabola at \(P .\) The parabola's focus lies at \(F(p, 0) .\) The ray \(L^{\prime}\) extending from \(P\) to the right is parallel to the \(x\) -axis. We show that light from \(F\) to \(P\) will be reflected out along \(L^{\prime}\) by showing that \(\beta\) equals \(\alpha .\) Establish this equality by taking the following steps. $$ \begin{array}{l}{\text { a. Show that } \tan \beta=2 p / y_{0}} \\ {\text { b. Show that } \tan \phi=y_{0} /\left(x_{0}-p\right)} \\ {\text { c. Use the identity }}\end{array} $$ $$ \tan \alpha=\frac{\tan \phi-\tan \beta}{1+\tan \phi \tan \beta} $$ \(\begin{aligned} & \text { to show that } \tan \alpha=2 p / y_{0} \\ \text { since } \alpha \text { and } \beta & \text { are both acute, tan } \beta=\tan \alpha \text { implies } \beta=\alpha \end{aligned}\) $$ \begin{array}{l}{\text { This reflective property of parabolas is used in applications like }} \\ {\text { car headlights, radio telescopes, and satellite TV dishes. }}\end{array} $$