Difference between revisions of "Ellipse"

Latest revision as of 20:52, 1 May 2021

An ellipse is a type of conic section.

Definition

An ellipse is formed by cutting through a cone at an angle. Equivalently, it is defined as the locus, or set, of all points $P$ such that the sum of the distances from $P$ to two fixed foci is a constant. (The equivalence of these two definitions is a non-trivial fact.)

Intuition

Ellipses tend to resemble circles which have been "flattened" or "stretched." They occur in nature as well as in mathematics: as was proven in Kepler's Laws, the planets all revolve about the sun in elliptical, not circular, orbits with the sun at one of the foci. Note that the circle is just a special case of the ellipse, just as a square is to a rectangle, and occurs when (in the first definition) the cutting plane is perpendicular to the axis of the the cone, or (in the second definition) the two foci of the ellipse coincide.

Using the second definition of an ellipse given above, one may easily construct an ellipse from household materials. To draw an ellipse with two pushpins, a loop of string, pencil, and paper, stick the pushpins in the paper place the string on the paper so that both pushpins are inside it. The pushpins will be the foci of the ellipse, and the length of the string will determine the sum of the distances from a point on the ellipse to the two foci. Hold the pencil on the paper such that the string is taut against the pencil tip and the two pushpins. Then move the pencil tip while keeping the string taut. This will trace out an ellipse.

Related Terminology

For a given non-circular ellipse, there will be two points on the ellipse closest to the center and two points furthest away -- it will be "tall and skinny" or "short and fat." The segment connecting the center of the ellipse to one of the "farther away ends" is called the semimajor axis and the segment connecting the center to a closer end is called the semiminor axis. These two segments are perpendicular. Drawing all four semi-axes divides the ellipse into 4 congruent quarters.

$[asy] size(200); D((-5,0)--(0,0)--(0,-3)); MC(90,"\mbox{semiminor axis}",7,D((0,0)--(0,3),green+linewidth(1)),E); MC("\mbox{semimajor axis}",7,D((0,0)--(5,0),red+linewidth(1)),S); D(ellipse(D("\mbox{center}",7,(0,0),black,SW),5,3),orange+linewidth(1)); [/asy]$

Ellipse

An ellipse in a Cartesian coordinate system with center $C = (h, k)$ whose axes are parallel to the coordinate axes, with the horizontal semi-axis of length $a$ and the vertical semi-axis of length $b$ is given by the equation $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$ . In particular, if the center of the ellipse is the origin this simplifies to $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ .

The three-dimensional counterpart of the ellipse is the ellipsoid.

Properties

Let $F_{1},F_{2}$ be the foci of an ellipse, let $P$ be a point on the ellipse, and let $\ell$ be the tangent line to the ellipse at $P$ . Then it is true that the bisector of angle $F_{1}PF_{2}$ is perpendicular to $\ell$ .

$[asy] size(200); defaultpen(fontsize(8)); pair P=(3,12/5), F1=(-4,0), F2=(4,0); D(ellipse((0,0),5,3)); D((-1,21/5)--(7,3/5)); D(F1--P--F2); D(P--P-(9/5,4)); dot(P^^F1^^F2); label("$P$",P,(1,1));label("$F_{1}$",F1,(0,-2));label("$F_{2}$",F2,(0,-2));label("$\ell$",(7,3/5),(0,2)); [/asy]$

Related Formulae

The area of an ellipse with semimajor and semiminor axes $a,b$ is $ab\pi$ .
The circumference of an ellipse is $4aE(\epsilon)$ , where the $E$ is the second elliptic integral.

Problems

Introductory

The ellipse with axis lengths $14$ and $16$ has the general equation of $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ . Find the value of $a^2+b^2$ . You do not need to find x or y.

Intermediate

Let $w_1$ and $w_2$ denote the circles $x^2+y^2+10x-24y-87=0$ and $x^2 +y^2-10x-24y+153=0,$ respectively. Let $m$ be the smallest possible value of $a$ for which the line $y=ax$ contains the center of a circle that is externally tangent to $w_2$ and internally tangent to $w_1.$ Given that $m^2=\frac pq,$ where $p$ and $q$ are relatively prime integers, find $p+q.$ (Source)
An equilateral triangle is inscribed in the ellipse whose equation is $x^2+4y^2=4$ . One vertex of the triangle is $(0,1)$ , one altitude is contained in the y-axis, and the length of each side is $\sqrt{\frac mn}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ . (Source)
An ellipse has foci at $(9,20)$ and $(49,55)$ in the $xy$ -plane and is tangent to the $x$ -axis. What is the length of its major axis? (Source)

@@ Line 5: / Line 5: @@
 Equivalently, it is defined as the [[locus]], or [[set]], of all [[point]]s <math>P</math> such that the sum of the distances from <math>P</math> to two fixed [[focus|foci]] is a constant.  (The equivalence of these two definitions is a non-trivial fact.)
-Ellipses tend to resemble [[circle]]s which have been "flattened" or "stretched."  They occur in nature as well as in mathematics: as was proven in [[Kepler's Laws]], the planets all revolve about the sun in elliptical, not circular, orbits with the sun at one of the foci.  Note that the circle is just a special case of the ellipse, just as a square is to a rectangle,  and occurs when (in the first definition) the cut is [[perpendicular]] to the axis of the the cone, or (in the second definition) the two foci of the ellipse coincide.
+==Intuition==
+Ellipses tend to resemble [[circle]]s which have been "flattened" or "stretched."  They occur in nature as well as in mathematics: as was proven in [[Kepler's Laws]], the planets all revolve about the sun in elliptical, not circular, orbits with the sun at one of the foci.  Note that the circle is just a special case of the ellipse, just as a square is to a rectangle,  and occurs when (in the first definition) the cutting plane is [[perpendicular]] to the axis of the the cone, or (in the second definition) the two foci of the ellipse coincide.
+Using the second definition of an ellipse given above, one may easily construct an ellipse from household materials. To draw an ellipse with two pushpins, a loop of string, pencil, and paper, stick the pushpins in the paper place the string on the paper so that both pushpins are inside it.  The pushpins will be the foci of the ellipse, and the length of the string will determine the sum of the distances from a point on the ellipse to the two foci.  Hold the pencil on the paper such that the string is taut against the pencil tip and the two pushpins. Then move the pencil tip while keeping the string taut. This will trace out an ellipse.
+==Related Terminology==
 For a given non-circular ellipse, there will be two points on the ellipse closest to the center and two points furthest away -- it will be "tall and skinny" or "short and fat."  The segment connecting the center of the ellipse to one of the "farther away ends" is called the ''[[semimajor axis]]'' and the segment connecting the center to a closer end is called the ''[[semiminor axis]]''.  These two segments are perpendicular.  Drawing all four semi-axes divides the ellipse into 4 [[congruent (geometry)|congruent]] quarters.
-{{image}}
+{{asy image|
+<asy>
+size(200);
+D((-5,0)--(0,0)--(0,-3));
+MC(90,"\mbox{semiminor axis}",7,D((0,0)--(0,3),green+linewidth(1)),E);
+MC("\mbox{semimajor axis}",7,D((0,0)--(5,0),red+linewidth(1)),S);
+D(ellipse(D("\mbox{center}",7,(0,0),black,SW),5,3),orange+linewidth(1));
+</asy>|right|Ellipse
+}}
-Using the second definition of an ellipse given above, one may easily construct an ellipse from household materials. To draw an ellipse with two pushpins, a loop of string, pencil, and paper, stick the pushpins in the paper place the string on the paper so that both pushpins are inside it.  The pushpins will be the foci of the ellipse, and the length of the string will determine the sum of the distances from a point on the ellipse to the two foci.  Hold the pencil on the paper such that the string is taut against the pencil tip and the two pushpins. Then move the pencil tip while keeping the string taut. This will traces out an ellipse.
+An ellipse in a [[Cartesian coordinate system]] with center <math>C = (h, k)</math> whose axes are parallel to the coordinate axes, with the horizontal semi-axis of length <math>a</math> and the vertical semi-axis of length <math>b</math> is given by the equation <math>\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1</math>.  In particular, if the center of the ellipse is the origin this simplifies to <math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1</math>.
-An ellipse in a [[Cartesian coordinate system]] with center <math>C = (h, k)</math> whose axes are parallel to the coordinate axes, with the vertical semi-axis of length <math>a</math> and the horizontal semi-axis of length <math>b</math> is given by the equation <math>\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1</math>.  In particular, if the center of the ellipse is the origin this simplifies to <math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1</math>.
+The three-dimensional counterpart of the ellipse is the [[ellipsoid]].
-The three-dimensional counterpart of the ellipse is the [[ellipsoid]].
+==Properties==
+* Let <math>F_{1},F_{2}</math> be the foci of an ellipse, let <math>P</math> be a point on the ellipse, and let <math>\ell</math> be the tangent line to the ellipse at <math>P</math>. Then it is true that the bisector of angle <math>F_{1}PF_{2}</math> is perpendicular to <math>\ell</math>.
+<asy>
+size(200);
+defaultpen(fontsize(8));
+pair P=(3,12/5), F1=(-4,0), F2=(4,0);
+D(ellipse((0,0),5,3));
+D((-1,21/5)--(7,3/5));
+D(F1--P--F2);
+D(P--P-(9/5,4));
+dot(P^^F1^^F2);
+label("$P$",P,(1,1));label("$F_{1}$",F1,(0,-2));label("$F_{2}$",F2,(0,-2));label("$\ell$",(7,3/5),(0,2));
+</asy>
 ==Related Formulae==
-*The [[area]] of an ellipse with major and minor axes <math>a,b</math> is <math>ab\pi</math>.
+*The [[area]] of an ellipse with semimajor and semiminor axes <math>a,b</math> is <math>ab\pi</math>.
 *The [[circumference]] of an ellipse is <math>4aE(\epsilon)</math>, where the <math>E</math> is the second [[elliptic integral]].
 ==Problems==
 ===Introductory===
+*The ellipse with axis lengths <math>14</math> and <math>16</math> has the general equation of <math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1</math>.  Find the value of <math>a^2+b^2</math>. You do not need to find x or y.
 ===Intermediate===
 *Let <math> w_1 </math> and <math> w_2 </math> denote the circles <math> x^2+y^2+10x-24y-87=0 </math> and <math> x^2 +y^2-10x-24y+153=0, </math> respectively. Let <math> m </math> be the smallest possible value of <math> a </math> for which the line <math> y=ax </math> contains the center of a circle that is externally tangent to <math> w_2 </math> and internally tangent to <math> w_1. </math> Given that <math> m^2=\frac pq, </math> where <math> p </math> and <math> q </math> are relatively prime integers, find <math> p+q. </math> ([[2005 AIME II Problems/Problem 15|Source]])

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