Ellipse
Ellipse is a closed curve that results from a cross section of the cone by a plane that does not pass through the apex. It looks like a flattened circle.
An ellipse is a symmetrical shape which is symmetric about horizontal and vertical axes.
The distance a+b is always constant for any point lying on the ellipse. Here $F_1$ and $F_2$ are the two foci of the ellipse. If any point on the ellipse is taken then the sum of the distances of the point from these two points is always constant. The size of the ellipse is determined by this sum of the two distances.
In coordinate geometry we can write the equation of an ellipse with center (h, k) as
$\frac{(x-h)^2}{a^2}$ + $\frac{(y-k)^2}{b^2}$ = 1
where a = half the length of the major axis
b = half the length of the minor axis.a>b
Ellipse is horizontal if the major is horizontal.
The ellipse with vertical major axis:
$\frac{(x-h)^2}{b^2}$ + $\frac{(y-k)^2}{a^2}$ = 1
a>b
The distance a+b is always constant for any point lying on the ellipse. Here $F_1$ and $F_2$ are the two foci of the ellipse. If any point on the ellipse is taken then the sum of the distances of the point from these two points is always constant. The size of the ellipse is determined by this sum of the two distances.
In coordinate geometry we can write the equation of an ellipse with center (h, k) as
$\frac{(x-h)^2}{a^2}$ + $\frac{(y-k)^2}{b^2}$ = 1
where a = half the length of the major axis
b = half the length of the minor axis.a>b
Ellipse is horizontal if the major is horizontal.
The ellipse with vertical major axis:
$\frac{(x-h)^2}{b^2}$ + $\frac{(y-k)^2}{a^2}$ = 1
a>b
Center of the ellipse is the midpoint of the segment joining the two foci. It is also the point of intersection of major and minor axes. In the above figure point C is the center of the ellipse.
Major axis is the longest diameter of the ellipse. It also equals the sum of the two distances of any point on the ellipse from the two focus points. An ellipse is always symmetrical about its major axis.
The length of the major axis =2a where a = distance of the endpoint of the major axis from the center.
The length of the major axis =2a where a = distance of the endpoint of the major axis from the center.
The shortest diameter of the ellipse is called as the minor axis. The length of the minor axis =2b where b is the distance of the endpoint of a minor axis from the center of the ellipse.
Foci are the two points lying on the major axis of the ellipse which are equidistant from the center. The positions of the foci decide the shape of the ellipse. If the two foci coincide then the ellipse becomes a circle.
The coordinates of the foci are given by
c²= a² - b² where c is the distance from the focus to center and b is the distance from the center to a co-vertex on the minor axis.
So the coordinates of the foci are given as (h ± c, k) when major axis is horizontal
F = (h, k ± c) if the major axis is vertical.
The coordinates of the foci are given by
c²= a² - b² where c is the distance from the focus to center and b is the distance from the center to a co-vertex on the minor axis.
So the coordinates of the foci are given as (h ± c, k) when major axis is horizontal
F = (h, k ± c) if the major axis is vertical.
Any line segment joining the two points lying on the ellipse and not passing through the center is called as a chord of the ellipse.
Secant is the line that intersects the ellipse at two points.
A line that touches the ellipse at only one point is called as tangent to the ellipse.
There are various ways to draw an ellipse.
1. Pin and String: This is the most easy method to draw an ellipse. Place a pin in each focus leaving a string with length 2a.Pull the string tightly with a pencil point and slide the pencil to draw the ellipse.
2. Trammel method: This is one of the easiest and accurate method to draw an ellipse. Mark a piece of paper with points O, A and B where OB = b, OA = a. Draw the two axes. Slide A along the minor axis and B along the major axis to get points. Point O traces the ellipse.
3. Foci and axes: When the major and minor axes are given foci can be determined. Foci lies on the major axis at distance a from the endpoint of the minor axis. Draw the arcs with radii a from the endpoints of the minor axes and make a smooth curve with endpoints of both the axes and the points obtained from the arcs.
4. Drawing an ellipse with two circles:
Step 1: Draw a the major axis and divide it into 3 equal parts and name it A, B, C and D.
Step 2: Take the radius as AB and draw 2 circles using B and C as centres. These 2 circles will be crossed each other in 2 points name them as E1 and E2.
Step 3: Draw 2 arcs on each circle by using the compass, with same radius, and by using A and D as centers. Name them as F1, F2, F3 and F4.
Step 4: Take the radius the length between F1 and E2 and run it from F1 to F2 using E2 as center. Repeat this step and draw the line from F3 to F4 by using E1 as center.
Finally make a smooth ellipse.
1. Pin and String: This is the most easy method to draw an ellipse. Place a pin in each focus leaving a string with length 2a.Pull the string tightly with a pencil point and slide the pencil to draw the ellipse.
2. Trammel method: This is one of the easiest and accurate method to draw an ellipse. Mark a piece of paper with points O, A and B where OB = b, OA = a. Draw the two axes. Slide A along the minor axis and B along the major axis to get points. Point O traces the ellipse.
3. Foci and axes: When the major and minor axes are given foci can be determined. Foci lies on the major axis at distance a from the endpoint of the minor axis. Draw the arcs with radii a from the endpoints of the minor axes and make a smooth curve with endpoints of both the axes and the points obtained from the arcs.
4. Drawing an ellipse with two circles:
Step 1: Draw a the major axis and divide it into 3 equal parts and name it A, B, C and D.
Step 2: Take the radius as AB and draw 2 circles using B and C as centres. These 2 circles will be crossed each other in 2 points name them as E1 and E2.
Step 3: Draw 2 arcs on each circle by using the compass, with same radius, and by using A and D as centers. Name them as F1, F2, F3 and F4.
Step 4: Take the radius the length between F1 and E2 and run it from F1 to F2 using E2 as center. Repeat this step and draw the line from F3 to F4 by using E1 as center.
Finally make a smooth ellipse.
Solved Examples
Question 1: Write the standard form of the equation for the ellipse shown below:
Solution:
Center of the ellipse is origin so (0, 0)
Major axis is from (-3, 0) to (3, 0)
So a = 3 ($\frac{Half\ of\ the\ measure\ axis}{Distance\ of\ one\ vertex\ from\ center}$)
Minor axis is from (0, -2) to (0, 2)
So b = 2 (half the length of minor axis)
So the equation is
$\frac{(x-h)^2}{a^2}$ + $\frac{(y-k)^2}{b^2}$ = 1
$\frac{(x)^2}{3^2}$ + $\frac{(y)^2}{2^2}$ = 1
$\frac{x^2}{9}$ + $\frac{(y)^2}{4}$ = 1
This ellipse has horizontal major axis.
Solution:
Center of the ellipse is origin so (0, 0)
Major axis is from (-3, 0) to (3, 0)
So a = 3 ($\frac{Half\ of\ the\ measure\ axis}{Distance\ of\ one\ vertex\ from\ center}$)
Minor axis is from (0, -2) to (0, 2)
So b = 2 (half the length of minor axis)
So the equation is
$\frac{(x-h)^2}{a^2}$ + $\frac{(y-k)^2}{b^2}$ = 1
$\frac{(x)^2}{3^2}$ + $\frac{(y)^2}{2^2}$ = 1
$\frac{x^2}{9}$ + $\frac{(y)^2}{4}$ = 1
This ellipse has horizontal major axis.
Question 2: Write the equation for the given ellipse.
Solution:
Center of the ellipse is origin so (0, 0) and it has vertical major axis.
Major axis is from (0, -4) to (0, 4)
So a =4 ($\frac{Half\ of\ the\ measure\ axis}{Distance\ of\ one\ vertex\ from\ center}$)
Minor axis is from (-3, 0) to (3, 0)
So b = 3 (half the length of minor axis)
So the equation is
$\frac{(x-h)^2}{b^2}$ + $\frac{(y-k)^2}{a^2}$ = 1
$\frac{(x)^2}{3^2}$ + $\frac{(y)^2}{4^2}$ = 1
$\frac{x^2}{9}$ + $\frac{(y)^2}{16}$ = 1
Solution:
Center of the ellipse is origin so (0, 0) and it has vertical major axis.
Major axis is from (0, -4) to (0, 4)
So a =4 ($\frac{Half\ of\ the\ measure\ axis}{Distance\ of\ one\ vertex\ from\ center}$)
Minor axis is from (-3, 0) to (3, 0)
So b = 3 (half the length of minor axis)
So the equation is
$\frac{(x-h)^2}{b^2}$ + $\frac{(y-k)^2}{a^2}$ = 1
$\frac{(x)^2}{3^2}$ + $\frac{(y)^2}{4^2}$ = 1
$\frac{x^2}{9}$ + $\frac{(y)^2}{16}$ = 1