[math] \newcommand{\ex}[1]{\item } \newcommand{\sx}{\item} \newcommand{\x}{\sx} \newcommand{\sxlab}[1]{} \newcommand{\xlab}{\sxlab} \newcommand{\prov}[1] {\quad #1} \newcommand{\provx}[1] {\quad \mbox{#1}} \newcommand{\intext}[1]{\quad \mbox{#1} \quad} \newcommand{\R}{\mathrm{\bf R}} \newcommand{\Q}{\mathrm{\bf Q}} \newcommand{\Z}{\mathrm{\bf Z}} \newcommand{\C}{\mathrm{\bf C}} \newcommand{\dt}{\textbf} \newcommand{\goesto}{\rightarrow} \newcommand{\ddxof}[1]{\frac{d #1}{d x}} \newcommand{\ddx}{\frac{d}{dx}} \newcommand{\ddt}{\frac{d}{dt}} \newcommand{\dydx}{\ddxof y} \newcommand{\nxder}[3]{\frac{d^{#1}{#2}}{d{#3}^{#1}}} \newcommand{\deriv}[2]{\frac{d^{#1}{#2}}{dx^{#1}}} \newcommand{\dist}{\mathrm{distance}} \newcommand{\arccot}{\mathrm{arccot\:}} \newcommand{\arccsc}{\mathrm{arccsc\:}} \newcommand{\arcsec}{\mathrm{arcsec\:}} \newcommand{\arctanh}{\mathrm{arctanh\:}} \newcommand{\arcsinh}{\mathrm{arcsinh\:}} \newcommand{\arccosh}{\mathrm{arccosh\:}} \newcommand{\sech}{\mathrm{sech\:}} \newcommand{\csch}{\mathrm{csch\:}} \newcommand{\conj}[1]{\overline{#1}} \newcommand{\mathds}{\mathbb} [/math]

It has been shown that the distance between a point on the hyperbola [math]\frac{x^2}{a^2} - \frac{y^2}{c^2-a^2} = 1[/math] and the focus [math](c,0)[/math] is [math]\left| \frac ca x - a \right|[/math]. Call this distance [math]d_1[/math].

• Show that the distance [math]d_2[/math] between a point on the hyperbola and the focus [math](-c,0)[/math] is [math]\left| \frac ca x + a \right|[/math].
• Show that [math]x \geq a[/math] for a point on the right branch of the hyperbola and that for such a point [math]d_1 = \frac ca x - a[/math] and [math]d_2 = \frac ca x + a[/math].
• Show that [math]x \leq -a[/math] for a point on the left branch of the hyperbola and that for such a point [math]d_1 = -\frac ca x + a[/math] and [math]d_2 = -\frac ca x - a[/math].
• Hence show that the graph of [math]\frac{x^2}{a^2} - \frac{y^2}{c^2-a^2} = 1[/math] contains only those points which satisfy the locus definition of hyperbola.