# Chapter 42: Chapter 42

Problem 746

The equation of an ellipse is $\mathrm{x}^{2} / \mathrm{a}^{2}+\mathrm{y}^{2} / \mathrm{b}^{2}=1\(. Discuss what happens if \)a=b=r$.

Problem 747

By definition, if an ellipse has a foci \((-c, 0)\) and \(F_{2}(c, 0)\), and \(\mathrm{P}(\mathrm{x}, \mathrm{y})\) is a point on the ellipse, then \(\mathrm{PF}_{1}+\mathrm{PF}_{2}=\mathrm{k}\), where \(\mathrm{k}\) is a constant such that $\mathrm{k}>\mathrm{F}_{1} \mathrm{~F}_{2}=2 \mathrm{c}\(. Assuming that the above holds, and defining a constant \)b$ such that \(b^{2}=a^{2}-c^{2}\), and a constant as such that $\mathrm{a}=\mathrm{k} / 2\(, prove that the equation of the ellipse is \)\mathrm{x}^{2} / \mathrm{a}^{2}+\mathrm{y}^{2} / \mathrm{b}^{2}=1$.

Problem 748

A single-lane highway must pass under a series of bridges. It is proposed that the bridges be shaped as semi-ellipses with the height equal to the width. The builder feels he must allow room for a 6 foot wide, 12 foot high truck to pass under it. What is the lowest bridge that can be built to serve this purpose.

Problem 749

The latus rectum of an ellipse is the chord through either Focus perpendicular to the major axis. Show that the length of the latus recta of ellipse $x^{2} / a^{2}+y^{2} / b^{2}=1\( is given by the formula \)\left(2 b^{2}\right) / a$.

Problem 750

Consider a point \(P_{1}\left(x_{1}, y_{1}\right)\) on the ellipse $b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}\( A tangent to the ellipse at \)p_{1}$ is a line through \(p_{1}\) with no other point on the ellipse. Prove that if $y_{1} \neq 0\(, there is a tangent at \)\mathrm{p}_{1}$, its slope is $\mathrm{m}=\left(-\mathrm{b}^{2} \mathrm{x}_{1}\right) /\left(\mathrm{a}^{2} \mathrm{y}_{1}\right)$ and its equation can be put in the form $\mathrm{x}_{1} \mathrm{x} / \mathrm{a}^{2}+\mathrm{y}_{1} \mathrm{y} / \mathrm{b}^{2}=1$.

Problem 751

Find the area of the ellipses (a) \(\mathrm{x}^{2} / 9+\mathrm{y}^{2} / 25=1\) (b) \(x^{2} / 144+y^{2} / 256=1\) (c) \(x^{2} / 64+y^{2} / 49=1\) (d) \(\mathrm{x}^{2} / 81+\mathrm{y}^{2} / 16=1\)