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Conic sections and their equations in the coordinate system
Ellipse
DEFINITION: An ellipse is the set of points in the plane the distance-sum of which measured from two given points of the plane is a constant greater than the distance of the two given points (Figure 72).
\latex{\overrightarrow v}
\latex{\frac 13 \overrightarrow v}
\latex{-2 \overrightarrow v}
\latex{ O }
\latex{ a }
\latex{ a }
\latex{ B }
\latex{ A }
\latex{ F_1 }
\latex{ a }
\latex{ c }
\latex{ b }
\latex{ C }
\latex{ D }
\latex{ r_1 }
\latex{ r_2 }
\latex{ P }
\latex{ a }
\latex{ c }
\latex{ F_2 }
\latex{ b }
Figure 72
The two given points \latex{(F_1, F_2)} are the two foci (plural of focus) of the ellipse.
The midpoint \latex{ O } of the line segment \latex{F_1F_2} is the centre of the ellipse.
The midpoint \latex{ O } of the line segment \latex{F_1F_2} is the centre of the ellipse.
The distance-sum \latex{F_1P + F_2P} that is constant in terms of any point \latex{ P } of the ellipse is usually denoted by \latex{ 2a }:
\latex{F_1P + F_2P = 2a}.
Resulting from the definition the ellipse is axially symmetric about the straight line \latex{F_1F_2} and also about the straight line perpendicular to \latex{F_1F_2} and intersecting it at \latex{ O }. It implies that the ellipse is centrally symmetric about the centre \latex{ O }.
Based on the definition the distance of the points \latex{ A } and \latex{ B } of the ellipse lying on the straight line \latex{F_1F_2} is
\latex{AB = 2a}.
The line segment \latex{ AB } with a length of \latex{ 2a } is the major axis of the ellipse.
The line segment defined by the points \latex{ C } and \latex{ D } of the perpendicular bisector of the major axis passing through \latex{ O } and also belonging to the ellipse is the minor axis of the ellipse; its length is denoted by \latex{2b}:
\latex{CD = 2b}.
\latex{F_1P} and \latex{F_2P} are the focal radii belonging to the point \latex{ P }. Usual notation:
\latex{F_1P=r_1} and \latex{F_2P=r_2}.
The distance \latex{F_1O = F_2O} is usually denoted by \latex{ c }.
Based on the definition with the notations introduced
\latex{AO = BO = F_1C = F_2C = F_2D = F_1D = a},
\latex{CO = DO = b},
\latex{F_1O = F_2O = c}.
\latex{CO = DO = b},
\latex{F_1O = F_2O = c}.
The ellipse is unambiguously defined by \latex{F_1}, \latex{F_2} and \latex{2a (\gt F_1F_2)}.
In Figure 72 it can be seen that the parameters \latex{ a }, \latex{ b }, \latex{ c } are not independent of each other, since for example by applying the Pythagorean theorem for the right-angled triangle \latex{F_1OC}
\latex{a^2 = b^2 + c^2}.
Example 1
Let us construct a few points of an ellipse given by its foci and the length of its major axis.
\latex{ O }
\latex{ F_2 }
\latex{ B }
\latex{ F_1 }
\latex{ A }
\latex{ a }
\latex{ a }
Figure 73
Solution
So \latex{F_1}, \latex{F_2} and a line segment with the length \latex{2a} are given so that \latex{2a \gt F_1F_2}.
If \latex{ O } is the midpoint of the line segment \latex{F_1F_2}, then the points \latex{ A } and \latex{ B } of the ellipse lying on the straight line of the major axis can be constructed easily by measuring half of the given line segment, i.e. a from \latex{ O } into both directions (Figure 73).
\latex{ O }
\latex{ F_2 }
\latex{ F_1 }
\latex{ A }
\latex{ B }
\latex{ C }
\latex{ D }
\latex{ a }
\latex{ a }
\latex{ a }
\latex{ a }
Figure 75
The two end-points of the minor axis are the points \latex{ C } and \latex{ D } of the perpendicular bisector of the line segment \latex{F_1F_2} for which \latex{F_1C = F_2C = F_2D = F_1D = a}. Thus the circle with the centre \latex{F_1} and with the radius a intersects the perpendicular bisector of \latex{F_1F_2} at the points \latex{ C } and \latex{ D } (Figure 74).
Two points of the ellipse, which do not lie on any of the axes of symmetry, can be obtained as follows:
\latex{ r_1 }
\latex{ r_2 }
\latex{ F_2 }
\latex{ P_2 }
\latex{ P_1 }
\latex{ B }
\latex{ A }
\latex{ C }
\latex{ D }
\latex{ P_4 }
\latex{ F_1 }
\latex{ P_3 }
Figure 75
- We divide the line segment with the given length \latex{ 2a } into two line segments \latex{r_1} and \latex{r_2} with different lengths.
- The intersection points of the circle with the centre \latex{F_1} and with the radius \latex{r_1} and the circle with the centre \latex{F_2} and with the radius \latex{r_2} are the points of the ellipse (Figure 75).
If we reflect the two points obtained about the straight line of the minor axis, two more points of the ellipse are obtained.
Finitely many points of the ellipse can be constructed with this process.
\latex{ 0 }
\latex{ x }
\latex{ y }
\latex{ b }
\latex{ a }
\latex{ c }
\latex{ r_1 }
\latex{ r_2 }
Figure 76
◆ ◆ ◆
Let us place the ellipse in the coordinate system so that its centre is the origin, and its foci are the points \latex{F_1(–c; 0)} and \latex{F_2(c; 0)}, where \latex{ c } is a positive real number given. Let also \latex{ a }, the length of half of the major axis be given \latex{(a \gt c)} (Figure 76).
The equation
The equation
\latex{\frac{x^2}{a^2}+\frac{y^2}{b^2}=1}
holds for the coordinates of the points \latex{P(x; y)} of the ellipse and only for these, where b is the length of half of the minor axis, i.e. \latex{b^2 = a^2 – c^2}.
The relation obtained is the general equation of the ellipse.
Example 2
Let us give the length of the major axis and the minor axis and also the foci of the ellipse with the origin as the centre, if its major axis lies on the \latex{ x }-axis, and its equation is \latex{16x^2 + 36y^2 – 576 = 0}.
Solution
As \latex{576 = 16 \times 36}, by dividing both sides of the equation given by \latex{ 576 } and by rearranging it, we obtain
\latex{\frac{x^2}{36}+\frac{y^2}{16}=1}, or \latex{\frac{x^2}{6^2}+\frac{y^2}{4^2}=1.}
By comparing it to the general form of the equation, we get\latex{ a = 6 }, \latex{ b = 4 }. So the length of the major axis is \latex{ 12 }, the length of the minor axis is \latex{ 8 }.
The foci lie on the \latex{ x }-axis symmetric about the origin. If using the usual notation \latex{ c } is their distance measured from the origin, then
The foci lie on the \latex{ x }-axis symmetric about the origin. If using the usual notation \latex{ c } is their distance measured from the origin, then
\latex{c=\sqrt{a^2-b^2}=\sqrt {20}=2\times\sqrt5\approx4.47}.
The foci are (Figure 77):
\latex{F_1(-2\times\sqrt5; 0)},
\latex{A(-6;0)}
\latex{C(0;4)}
\latex{P(x;y)}
\latex{B(6;0)}
\latex{D(0;-4)}
\latex{F_1(-2\times\sqrt5;0)}
\latex{F_2(2\times\sqrt5;0)}
\latex{x }
\latex{ y }
\latex{ 1 }
\latex{0 }
\latex{1 }
Figure 77
Hyperbola
DEFINITION: A hyperbola is the set of points in the plane the absolute value of the distance-difference of which measured from two given points of the plane is a constant less than the distance of the two given points (Figure 78).
The two given points \latex{F_1}, \latex{F_2} are the foci, the midpoint \latex{ O } of the line segment \latex{F_1F_2} is the centre of the hyperbola.
The line segments connecting an arbitrary point \latex{ P } of the hyperbola with the foci are the focal radii of \latex{P: F_1P = r_1,\;\; F_2P = r_2}.
The usual notation of the given distance \latex{\left|r_1-r_2\right|} is \latex{2a}:
The line segments connecting an arbitrary point \latex{ P } of the hyperbola with the foci are the focal radii of \latex{P: F_1P = r_1,\;\; F_2P = r_2}.
The usual notation of the given distance \latex{\left|r_1-r_2\right|} is \latex{2a}:
\latex{\lvert r_1-r_2\rvert=2a}.
The definition implies that a hyperbola is not a continuous point set, rather it consist of two parts, two branches.
\latex{ O }
\latex{ A }
\latex{ B }
\latex{ F_2 }
\latex{ r_2 }
\latex{ P }
\latex{ r_1 }
\latex{ C }
\latex{ D }
\latex{ F_1 }
\latex{ a }
\latex{ a }
\latex{ b }
\latex{ b }
\latex{ c }
\latex{ c }
Figure 78
Based on the definition it holds for the points \latex{ A } and \latex{ B } of the straight line \latex{F_1F_2} also belonging to the hyperbola that \latex{AB = 2a}. The line segment \latex{ AB } is the real axis of the hyperbola. (The straight line \latex{ AB } is also usually called the real axis.)
The definition implies that the hyperbola is axially symmetric about the straight line \latex{F_1F_2} and also about the straight line perpendicular to it and passing through the point \latex{ O }, hence it is centrally symmetric about \latex{ O }. The distance \latex{F_1O = F_2O} is usually denoted by \latex{ c }, thus \latex{F_1F_2 = 2c}.
If \latex{ C } and \latex{ D } denote the points of the perpendicular bisector of \latex{F_1F_2} for which \latex{AC = BC = AD = BD = c}, then the Pythagorean theorem implies that
\latex{OC^2 = OD^2 = c^2 – a^2}.
The length of the line segments \latex{ OC } and \latex{ OD } is usually denoted by \latex{ b }.
Thus
Thus
\latex{c^2 = a^2 + b^2}.
The line segment \latex{ CD } with the length \latex{ 2b } is the imaginary axis of the hyperbola. (The straight line \latex{ CD } is also usually called the imaginary axis.)
Let us place the hyperbola in the coordinate system so that its centre is the origin, and its foci are the points \latex{F_1(–c; 0)} and \latex{F_2(c; 0)}, where \latex{ c } is a positive real number given. \latex{ a }, the length of half of the real axis is also given so that \latex{c \gt a} (Figure 79).
The equation
Let us place the hyperbola in the coordinate system so that its centre is the origin, and its foci are the points \latex{F_1(–c; 0)} and \latex{F_2(c; 0)}, where \latex{ c } is a positive real number given. \latex{ a }, the length of half of the real axis is also given so that \latex{c \gt a} (Figure 79).
The equation
\latex{\frac{x^2}{a^2}-\frac{y^2}{b^2}=1}
holds for the coordinates of the points \latex{P(x; y)} of the hyperbola and only for these, where \latex{ b } is the length of half of the imaginary axis, i.e. \latex{b^2 = c^2 – a^2}.
This equation is the general equation of the hyperbola.
This equation is the general equation of the hyperbola.
\latex{P(x; y)}
\latex{F_2(c; 0)}
\latex{F_1(-c; 0)}
\latex{ O }
\latex{ x }
\latex{ a }
\latex{ a }
\latex{ y }
\latex{ r_1 }
\latex{ r_2 }
Figure 79
The circle, the parabola, the ellipse and the hyperbola
as conic sections
as conic sections
Although their shapes are substantially different, the equation of a circle, a parabola, an ellipse and also a hyperbola in the planar Cartesian coordinate system is a quadratic equation in two variables. We can also see some relationship between the spatial origins of these curves. All of them can be generated as the plane sections of a right circular cone (double right circular cone in the case of a hyperbola) and certain planes.
It can be proven that the curves generated as lines of intersection this way meet the conditions formulated in the planar definitions.
Figure 80
Exercises
{{exercise_number}}. A point on a straight line is \latex{P_0(0; 2)}, a direction vector is \latex{\overrightarrow v (3; 1)}. Calculate the first coordinate of the point \latex{ P } on the straight line the second coordinate of which is
- \latex{9x^2+25y^2=225};
- \latex{x^2+35y^2=225};
- \latex{\frac{x^2}{5}+\frac{y^2}{3}=1}.
Also give the coordinates of the foci in each case.
{{exercise_number}}. Set up the equation of the ellipse if the length of its minor axis is \latex{ 32 }, its axes of symmetry are the coordinate axes and one of its foci is
- \latex{F_1(-3; 0)};
- \latex{F_1(5; 0)};
- \latex{F_1(-1; 0)};
- \latex{F_1(\sqrt7; 0)}.
{{exercise_number}}. The two foci and the length of the real axis of a hyperbola are given. Construct a few points of the hyperbola.
{{exercise_number}}. Set up the general equation of the hyperbola if its axes of symmetry are the coordinate axes, the length of its real axis is \latex{ 6 } and one of its foci is
- \latex{F_1(5; 0)};
- \latex{F_1(-7; 0)};
- \latex{F_1(9; 0)};
- \latex{F_1(-\sqrt{10}; 0)}.
{{exercise_number}}. Give the axes of symmetry and the foci of the graph of the following function:
\latex{f:\left(\R\backslash\left\{0\right\}\right)\rightarrow\R,\;\;f(x)=\frac{2}{x}}.