Marco Antonio Rodriguez Andrade
Master of Mathematics, Dr. of Science
The ellipse is one of the conics that has been studied since the time of the Greeks, on the reference of Apollonius, and that has been useful in explaining the trajectory of the planets and in the design of rooms with exceptional acoustic characteristics. When a right cone is intersected by a plane neither parallel to the generatrix of the cone nor perpendicular to its base, the result is an Ellipse, this fact. When the plane is parallel, a circle is formed and this case deserves special treatment.
Given two fixed points \({F_1}\) and \({F_2}\), which we will call the Foci of the Ellipse and a constant \(k > 0\); the Ellipse with foci at the points \({F_1}\) and \({F_2}\), is the locus of the points in the plane whose distances to the points \({F_1}\) and \({F_2 }\) is equal to a constant \(k\), that is:
\(P{F_1} + P{F_2} = k\)
Construction of the ellipse with ruler and compass
Given two fixed points \({F_1}\) , \({F_2}\) and a constant \(k > 0\) we can construct several points that belong to the ellipse that satisfies:
\(P{F_1} + P{F_2} = k;\)
To do this, it is enough to carry out the following steps.
Initial situation
On the point \({F_1}\) an arc of radius \({a_1}\) (green) is drawn and on \({F_2}\) a circle of radius \(k – {a_1}\) (blue). The intersections of these arcs are marked, in this case \({P_1}\) and \(P_1^\prime\).
The way in which the radii were chosen is guaranteed:
\({F_1}{P_1} + {F_2}{P_1} = k\)
\({F_1}P{^\prime_1} + {F_2}P{^\prime_1} = k\)
On the point \({F_2}\) an arc of radius \({a_1}\) (green) is drawn and on \({F_1}\) a circle of radius \(k – {a_1}\) (blue). The intersections of these arcs are marked, in this case \({Q_1}\) and \(Q_1^\prime\)
On the point \({F_1}\) an arc of radius \({a_2}\) (green) is drawn and on \({F_2}\) a circle of radius \(k – {a_2}\) (blue). The intersections of these arcs are marked, in this case \({P_2}\) and \(P_2^\prime\).
The way in which the radii were chosen is guaranteed:
\({F_1}{P_2} + {F_2}{P_2} = k\)
\({F_1}P{^\prime_2} + {F_2}P{^\prime_3} = k\)
On the point \({F_2}\) an arc of radius \({a_2}\) (green) is drawn and on \({F_1}\) a circle of radius \(k – {a_2}\) (blue). The intersections of these arcs are marked, in this case \({Q_2}\) and \(Q_2^\prime\)
Similarly, more points are constructed.
By joining the points, the outline of an ellipse is obtained whose foci are the points \({F_1}\) and \({F_2}.\)
Elements and characteristics of the ellipse
The following figure shows other important elements of the ellipse.
Element Description Example Center of Ellipse Midpoint of segment \(\overline {{F_1}{F_2}} \), where \({F_1}\) and \({F_2}\) are the foci of the ellipse. \(O\) Focal axis or major axis It is the line that passes through the foci Vertex of the ellipse Intersection of the focal axis with the ellipse \({V_1}\) and \({V_2}\) Conjugate axis or minor axis Segment of line that joins two points of the ellipse Chord Segment of line that joins two points of the ellipse \(\overline {P_3^\prime Q_1^\prime } \) Focal Chord Chord that passes through one of the foci of the ellipse \ (\overline {{P_1}S} \) Straight side Focal chord perpendicular to the focal axis \(\overline {RR^\prime } \) Diameter of the ellipse Chord passing through the center \(\overline {{P_1}P_3 ^\prime } \)
ellipse characteristics
The ellipse has two axes of symmetry namely: the focal axis and the conjugate axis we will denote with
\(a = O{V_1},\)\(b = OP\)\(c = O{F_1}\)
Element Description Value Semi Major Axis Length of Segment \(\overline {O{V_1}} \) \(a\) Semi Minor Axis Length of Segment \(\overline {OQ} \) \(b\) Major Axis Length of segment \(\overline {{V_1}{V_2}} \) \(2a\) Minor axis Segment length \(\overline {QQ’} \) \(2b\) Focal distance Segment length \(\overline { {F_1}{F_2}} \) 2\(c\)
They have the following:
\(k = 2a\)
That is to say:
\(P{F_1} + P{F_2} = k = 2a,\;\)
By the Pythagorean theorem:
\({b^2} + {c^2} = {a^2}.\)
One way to draw an ellipse in a design program is to determine the location of the center and the values of the major axis and the minor axis, and it is not necessary to indicate the position of each of the foci of the ellipse; which can be determined using the relationship:
\({b^2} + {c^2} = {a^2}.\)
For example, if an ellipse is drawn in a design program whose axes measure 6cm and 10cm, then the semi-axes measure 3cm and 5cm; and the focal length can be calculated as follows:
\({3^2} + {c^2} = {5^2}\)
Solving the equation, it is obtained that the focal length is 4 cm.
Equations that model ellipses
Given two fixed points \({F_1}\) and \({F_2}\), which we will call the Foci of the Ellipse and a constant \(a > 0\); the Ellipse with foci at the points \({F_1}\) and \({F_2}\), is the locus of the points in the plane whose distances to the points \({F_1}\) and \({F_2 }\) is equal to a constant \(k\), that is:
\(P{F_1} + P{F_2} = 2a\)
Equations of the ellipse with center at the origin and focal axis on one of the coordinate axes
Ellipse position
Center at \(\left( {0,0} \right).\) Foci at \({F_1}\left( { – c,0} \right),\;{F_2}\left( {c,0 } \right).\) Vertices at \({V_1}\left( { – a,0} \right),\;{V_2}\left( {a,0} \right).\;\)Focal Axis : X axis\)
parabola equation
\(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\)
Where it is fulfilled:
\({b^2} + {c^2} = {a^2}.\)
Ellipse position
Center at \(\left( {0,0} \right).\) Foci at \({F_1}\left( {0, – c} \right),\;{F_2}\left( {0,c } \right).\) Vertices at \({V_1}\left( {0, – a} \right),\;{V_2}\left( {0,a} \right).\;\)Focal Axis Axis y\)
parabola equation
\(\frac{{{x^2}}}{{{b^2}}} + \frac{{{y^2}}}{{{a^2}}} = 1\)
Where it is fulfilled:
\({b^2} + {c^2} = {a^2}\)
worked examples
1. Characteristics of the Ellipse
The focal axis is on the ordinate axis (the \(y\) axis) its semi-axes measure 7 and 2 respectively and its center is at the origin
Equation of the Ellipse
In this case
\(a = 7,\;b = 2,\) therefore
\(c = \sqrt {{7^2} – {2^2}} = \sqrt {45} = \sqrt {{3^2}5} = 3\sqrt 5 \)
\(\frac{{{x^2}}}{{{b^2}}} + \frac{{{y^2}}}{{{a^2}}} = 1\)
\(\frac{{{x^2}}}{{{2^2}}} + \frac{{{y^2}}}{{{7^2}}} = 1\)
\(\frac{{{x^2}}}{4} + \frac{{{y^2}}}{{49}} = 1\)
graph sketch
2. Characteristics of the Ellipse
The focal axis is on the abscissa axis (the \(x\) axis), its focal length measures 8, its major axis measures 10 and its center is at the origin
Equation of the Ellipse
In this case
\(a = 4,\;c = 5,\) therefore
\(b = \sqrt {{5^2} – {4^2}} = \sqrt 9 = 3\)
\(\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\)
\(\frac{{{x^2}}}{{{4^2}}} + \frac{{{y^2}}}{{{3^2}}} = 1\)
\(\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{9} = 1\)
graph sketch
Equations on an ellipse with the property of a center that is outside the origin, and with a focal axis parallel to one of the coordinate axes
Ellipse position
Center at \(\left( {h,k} \right).\) Foci at \({F_1}\left( {h – c,k} \right),\;{F_2}\left( {h + c,k} \right).\) Vertices in \({V_1}\left( {h – a,k} \right),\;{V_2}\left( {h + a,k} \right). \;\)Focal Axis: \(y = k\)
ellipse equation
\(\frac{{{{\left( {x – h} \right)}^2}}}{{{a^2}}} + \frac{{{{\left( {y – k} \ right)}^2}}}{{{b^2}}} = 1\)
Where it is fulfilled:
\({b^2} + {c^2} = {a^2}.\)
Ellipse position
Center at \(\left( {h,k} \right).\) Foci at \({F_1}\left( {h,k – c} \right),\;{F_2}\left( {h, k + c} \right).\) Vertices in \({V_1}\left( {h,k – a} \right),\;{V_2}\left( {h,k + a} \right). \;\)Focal Axis \(x = h\)
ellipse equation
\(\frac{{{{\left( {x – h} \right)}^2}}}{{{b^2}}} + \frac{{{{\left( {y – k} \ right)}^2}}}{{{a^2}}} = 1\)
Where it is fulfilled:
\({b^2} + {c^2} = {a^2}\)
worked examples
1. Characteristics of the Ellipse
The foci are \({F_1}\left( { – 10, – 3} \right),\;{F_2}\left( {14, – 3} \right)\) and their major axis is equal to 26.
Equation of the Ellipse
In this case: \(2c = {F_1}{F_2} = 24,\) therefore: \(c = 12\)
\(2a = 26\)
\(a = 13\)
\({b^2} + {c^2} = {a^2}\)
\({b^2} + {12^2} = {13^2}\)
\({b^2} = {13^2} – {12^2}\)
\({b^2} = 25\)
The center of the ellipse is at the midpoint of the segment \(\overline {{F_1}{F_2}} \) which is \(C\left( {2, – 3} \right).\)
The focal axis is parallel to the \(x\) axis.
\(\frac{{{{\left( {x – h} \right)}^2}}}{{{a^2}}} + \frac{{{{\left( {y – k} \ right)}^2}}}{{{b^2}}} = 1\)
\(\frac{{{{\left( {x – 2} \right)}^2}}}{{{{13}^2}}} + \frac{{{{\left( {y – \ left( { – 3} \right)} \right)}^2}}}{{{5^2}}} = 1\)
\(\frac{{{{\left( {x – 2} \right)}^2}}}{{25}} + \frac{{{{\left( {y + 3)} \right)} ^2}}}{{169}} = 1\)
graph sketch
2. Characteristics of the Ellipse
The foci are \({F_1}\left( {1,1} \right),\;{F_2}\left( {1, – 1} \right)\) and their minor axis is equal to 2.
Equation of the Ellipse
In this case: \(2c = {F_1}{F_2} = 2,\) therefore: \(c = 1.\)
\(2b = 2\)
\(b = 1\)
\({b^2} + {c^2} = {a^2}\)
\({1^2} + {1^2} = {a^2}\)
\(2 = {a^2}\)
The center of the ellipse is at the midpoint of the segment \(\overline {{F_1}{F_2}} \) which is \(C\left( {1,0} \right).\)
The focal axis is parallel to the \(y\) axis.
\(\frac{{{{\left( {x – h} \right)}^2}}}{{{a^2}}} + \frac{{{{\left( {y – k} \ right)}^2}}}{{{b^2}}} = 1\)
\(\frac{{{{\left( {x – 0} \right)}^2}}}{2} + \frac{{{{\left( {y – 1)} \right)}^2 }}}{{{1^2}}} = 1\)
\(\frac{{{x^2}}}{2} + {\left( {y – 1} \right)^2} = 1\)
graph sketch
Following