Directrix of conic
This line segment forms a right triangle with hypotenuse length \(a\) and leg lengths \(b\) and \(c\). The directrix of a conic section is the line which, together with the point known as the focus, serves to define a conic section as the locus of points whose distance from the focus is proportional to the horizontal distance from the directrix, with being the constant of proportionality. The equations of the directrices of a horizontal ellipse are \(x=±\dfrac{a^2}{c}\). Find the directrix and an equation for this parabola. If the directrix is given in terms of [latex]y[/latex], we use the general polar form in terms of sine. This allows a small light bulb to illuminate a wide angle of space in front of the flashlight or car.An ellipse can also be defined in terms of distances. Step 1: The distance from the vertex to the focus is 2 = d, the focal distance.
The value of Determine the eccentricity of the hyperbola described by the equationFirst find the values of a and b, then determine c using the equation \(c^2=a^2+b^2\).Sometimes it is useful to write or identify the equation of a conic section in polar form. The vertex of the right branch has coordinates \((a,0),\) soThis equation is therefore true for any point on the hyperbola. Download for free at \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)[ "article:topic", "directrix", "discriminant", "nappes", "focus", "Hyperbolas", "Ellipses", "Parabola", "CONIC SECTIONS", "major axis", "minor axis", "eccentricity", "vertex", "focal parameter", "general form", "nappe", "standard form", "license:ccbyncsa", "showtoc:no", "authorname:openstaxstrang" ][ "article:topic", "directrix", "discriminant", "nappes", "focus", "Hyperbolas", "Ellipses", "Parabola", "CONIC SECTIONS", "major axis", "minor axis", "eccentricity", "vertex", "focal parameter", "general form", "nappe", "standard form", "license:ccbyncsa", "showtoc:no", "authorname:openstaxstrang" ]\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)Example \(\PageIndex{1}\): Converting the Equation of a Parabola from General into Standard FormExample \(\PageIndex{2}\): Finding the Standard Form of an EllipseExample \(\PageIndex{3}\): Finding the Standard Form of a HyperbolaExample \(\PageIndex{4}\): Determining Eccentricity of a Conic SectionExample \(\PageIndex{5}\): Graphing a Conic Section in Polar CoordinatesExample \(\PageIndex{6}\): Identifying a Rotated Conic11.E: Parametric Equations and Polar Coordinates (Exercises) So that's (a,b) is the focus. Determine whether the directrix is horizontal or vertical. Since the first set of parentheses has a 9 in front, we are actually adding 36 to the left-hand side.
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The value of Determine the eccentricity of the hyperbola described by the equationFirst find the values of a and b, then determine c using the equation \(c^2=a^2+b^2\).Sometimes it is useful to write or identify the equation of a conic section in polar form. The vertex of the right branch has coordinates \((a,0),\) soThis equation is therefore true for any point on the hyperbola. Download for free at \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)[ "article:topic", "directrix", "discriminant", "nappes", "focus", "Hyperbolas", "Ellipses", "Parabola", "CONIC SECTIONS", "major axis", "minor axis", "eccentricity", "vertex", "focal parameter", "general form", "nappe", "standard form", "license:ccbyncsa", "showtoc:no", "authorname:openstaxstrang" ][ "article:topic", "directrix", "discriminant", "nappes", "focus", "Hyperbolas", "Ellipses", "Parabola", "CONIC SECTIONS", "major axis", "minor axis", "eccentricity", "vertex", "focal parameter", "general form", "nappe", "standard form", "license:ccbyncsa", "showtoc:no", "authorname:openstaxstrang" ]\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)Example \(\PageIndex{1}\): Converting the Equation of a Parabola from General into Standard FormExample \(\PageIndex{2}\): Finding the Standard Form of an EllipseExample \(\PageIndex{3}\): Finding the Standard Form of a HyperbolaExample \(\PageIndex{4}\): Determining Eccentricity of a Conic SectionExample \(\PageIndex{5}\): Graphing a Conic Section in Polar CoordinatesExample \(\PageIndex{6}\): Identifying a Rotated Conic11.E: Parametric Equations and Polar Coordinates (Exercises) So that's (a,b) is the focus. Determine whether the directrix is horizontal or vertical. Since the first set of parentheses has a 9 in front, we are actually adding 36 to the left-hand side.
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