Ellipse
Ellipse is conic section defined as a plane curve obtained by intersection of a cone and a plane in which
the plane misses the vertex and
the plane is NOT at right angle with the axis-
Ellipse is conic section defined as a plane curve obtained by intersection of a cone and a plane in which if
α= angle between generator and axis and
β= angle between plane and axis,
and
α < β < 90
then
the section is a ellipse,
in which
eccentricity = \(\frac{\cos \beta}{\cos \alpha}\)
here
the eccentricity is the measure of how far the conic deviates from being circular Ellipse is a plane curve defined a locus of a point in which
\( \frac{\text{distance from a fixed point}}{\text{distance from a fixed line}} \)= constant (e <1)In this definition of conic section,
the constant ratio is called eccentricity, it is denoted by e.
the fixed point is called focus.
the fixed line is called directrix.
- Ellipse is a plane curve defined a locus of a point in which the distance from a fixed point (or focus) is always less than the distance from a fixed line (or directrix)
-
Ellipse is a plane curve defined a locus of a point whose sum of distances from two fixed points (foci) is always a constant.
In this definition
- Foci: The two fixed points of ellipse
- Directrix: The two fixed line of ellipse
- Axis: The straight line passing through focus and perpendicular to directrix
- Major axis: The straight line passing through the foci
- Minor axis: The straight line passing through the center and perpendicular to the major axis
- Length of major axis: The distance between the vertices on major axis
- Length of minor axis: The distance between the meeting points of minor axis with ellipse
- Vertices: The meeting points of the major axis with ellipse
- Centre of ellipse: The middle point of the join of foci
- Latus rectum: The chord passing through focus and perpendicular to major axis
Please Note that
- Every ellipse has two axes of symmetry. The longer axis is called the major axis, and the shorter axis is called the minor axis.
- Each endpoint of the major axis is the vertex of the ellipse, and each endpoint of the minor axis is a co-vertex of the ellipse.
- The center of an ellipse is the midpoint of both the major and minor axes.
The axes are perpendicular at the center. - The foci always lie on the major axis
There are four variations of the standard form of the ellipse. These variations are categorized first by
the location of the center (the origin or not the origin),
and by
the position (horizontal or vertical).
Ellipse: Proof of Basic Facts
- c=ae
If Z and Z' are the directrix, F(c,0) and F'(-c,0) are the foci, and A(a,0) and A;(-a,0) are the vertices of ellipse, then by the definition of conic we have
\(\frac{A'F}{A'Z}=e\)
or \(A'F=A'Z e\) (1)
Similarly, we have
\(\frac{AF}{AZ}=e\)
or \(AF=AZ e\) (2)
Substracting (2) from (1), we get
\(A'F-AF=(A'Z-AZ) e\)
or\((A'O+OF)-(AO-OF)=(A'A) e\)
or\(2OF=2a e\)
or\(2c=2a e\)
or\(c=a e\) - sum of distance from Foci is 2a.
the distance of a point A(a,0) from fous F(c,0) is
a−(c)=a-c.
the distance of a point A(a,0) from fous F'(-c,0) is
a−(-c)=a+c
The sum of the distances from Foci is
(a+c)+(a−c)=2a - Relation between a, b, c.
\(2 \sqrt{b^2+c^2}=2a\)
or\(b^2+c^2=a^2\)
Standard Equation of Ellipse
Let C be an ellipse whose
foci are (−c,0) and (c,0).
center is O: (0,0)
Take any point P(x,y) on ellipse C,
If (a,0) is a vertex of the ellipse, then the distance from (−c,0) to (a,0) is
a−(−c)=a+c.
The distance from (c,0) to (a,0) is
a−c.
The sum of the distances from the foci to the vertex is
(a+c)+(a−c)=2a
If (x,y) is a point on the ellipse, then we can define the following variables:
d1=the distance from (−c,0)to (x,y)
d2=the distance from (c,0)to (x,y)
By the definition of an ellipse,
d1+d2=2a
Here
\(d_1+d_2=2a\)
or
\(\sqrt{(x+c)^2+y^2}+\sqrt{(x−c)^2+y^2}=2a\)
or
\(\sqrt{(x+c)^2+y^2}=2a-\sqrt{(x−c)^2+y^2}\)
or
\((x+c)^2+y^2=\left [2a-\sqrt{(x−c)^2+y^2} \right] ^2\)
or
\(x^2+2xc+c^2+y^2=4a^2-4a\sqrt{(x−c)^2+y^2} +(x−c)^2+y^2 \)
or
\(x^2+2xc+c^2+y^2=4a^2-4a\sqrt{(x−c)^2+y^2} +x^2-2xc+c^2+y^2 \)
or
\(2xc=4a^2-4a\sqrt{(x−c)^2+y^2} -2xc\)
or
\(4xc-4a^2=-4a\sqrt{(x−c)^2+y^2}\)
or
\(xc-a^2=-a\sqrt{(x−c)^2+y^2}\)
or
\(\left[xc-a^2\right]^2= \left[ -a\sqrt{(x−c)^2+y^2} \right ]^2\)
or
\(c^2x^2-2a^2cx+a^4= a^2 \left[(x−c)^2+y^2 \right ]\)
or
\(c^2x^2-2a^2cx+a^4= a^2 (x^2-2xc+c^2+y^2 ) \)
or
\(c^2x^2-2a^2cx+a^4= a^2x^2-a^22xc+a^2c^2+a^2y^2 \)
or
\(a^2x^2-c^2x^2+a^2y^2=a^4-a^2c^2 \)
or
\(x^2(a^2-c^2)+a^2y^2=a^2(a^2-c^2) \)
or
\(x^2b^2+a^2y^2=a^2b^2 \) \(a^2-c^2=b^2\)
or
\(\frac{x^2}{a^2}+ \frac{y^2}{b^2}=1 \)
Thus, the standard equation of an ellipse is
\(\frac{x^2}{a^2}+ \frac{y^2}{b^2}=1 \)
This equation defines an ellipse centered at the origin.
- If a>b, the ellipse is stretched further in the horizontal direction
- if b>a the ellipse is stretched further in the vertical direction.
- When c = 0, both foci merge together with the center of the ellipse and so the ellipse becomes a circle
- When c = a, then b = 0. The ellipse reduces to the line segment joining the two foci
Summary of parameters in an Ellipse
| Ellipse | Ellipse | Ellipse | Ellipse | |
| Equation | \( \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\); a > b | \( \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\); a < b | \( \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\); a > b | \( \frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\); a < b |
| Center | (0,0) | (0,0) | (h,k) | (h,k) |
| Vertex | \( (\pm a,0) \) | \( (0,\pm b) \) | \( (h\pm a,k) \) | \( (h,k\pm b) \) |
| Focus | \( (\pm ae,0) \) | \( (0,\pm be) \) | \( (h\pm ae,k) \) | \( (h,k\pm be) \) |
| Directrix | \( x=\pm \frac{a}{e} \) | \( y=\pm \frac{b}{e} \) | \( x=h\pm \frac{a}{e} \) | \( y=k\pm \frac{b}{e} \) |
| Length Rectum | \( 2 \frac{b^2}{a} \) | \( 2 \frac{a^2}{b}\) | \( 2 \frac{b^2}{a} \) | \( 2 \frac{a^2}{b}\) |
| Eccentricity | \( b^2=a^2(1-e^2) \) | \( a^2=b^2(1-e^2) \) | \( b^2=a^2(1-e^2) \) | \( a^2=b^2(1-e^2) \) |
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