Ellipse
Push two drawing pins into a board, tie the ends of a piece of string to each one, and draw with a pencil, keeping the string taut all the time: you are drawing an ellipse. It is the curve of planetary orbits (with the Sun at a focus, not at the centre) and of rooms with a "whispering effect" such as the Whispering Gallery.
The ellipse as a locus
Just as you would draw the ellipse with the string: drag P on the revision circle if you need to remember what "locus" means before going on.
Given the ellipse with equation $\frac{x^2}{25} + \frac{y^2}{9} = 1$, determine the coordinates of its vertices, those of its foci, and the value of its eccentricity.
Given the ellipse with equation $\frac{x^2}{25} + \frac{y^2}{9} = 1$, determine the coordinates of its vertices, those of its foci, and the value of its eccentricity.
Elements and standard equation
Write the standard equation of an ellipse centred at the origin of coordinates, knowing that one of its foci is the point $F(4, 0)$ and that the length of its major axis…
Write the standard equation of an ellipse centred at the origin of coordinates, knowing that one of its foci is the point $F(4, 0)$ and that the length of its major axis is $10$.
An ellipse has its centre at the point $C(-2, 3)$, its major axis is parallel to the x-axis and measures $12$ units, and its minor axis measures $8$ units. Write its…
An ellipse has its centre at the point $C(-2, 3)$, its major axis is parallel to the x-axis and measures $12$ units, and its minor axis measures $8$ units. Write its equation in canonical form.
Eccentricity: how flattened the ellipse is
Calculate the eccentricity of the ellipse whose equation is $16x^2 + 25y^2 = 400$.
Calculate the eccentricity of the ellipse whose equation is $16x^2 + 25y^2 = 400$.
The orbit of Mercury around the Sun is an ellipse with the Sun at one of its foci. The minimum distance from Mercury to the Sun (perihelion) is $46$ million kilometres,…
The orbit of Mercury around the Sun is an ellipse with the Sun at one of its foci. The minimum distance from Mercury to the Sun (perihelion) is $46$ million kilometres, and the maximum distance (aphelion) is $70$ million kilometres. Calculate the eccentricity of Mercury's orbit and the length of its semi-minor axis in million kilometres.
General equation and translated ellipses
If the centre is not at the origin, the standard equation is translated in the same way as that of the circle: replace $x$ with $x-x_0$ and $y$ with $y-y_0$.
$$\frac{(x-x_0)^2}{a^2}+\frac{(y-y_0)^2}{b^2}=1$$From the equation of the ellipse $\frac{(x-1)^2}{16} + \frac{(y+4)^2}{9} = 1$, state the coordinates of its centre and the lengths of its semi-axes.
From the equation of the ellipse $\frac{(x-1)^2}{16} + \frac{(y+4)^2}{9} = 1$, state the coordinates of its centre and the lengths of its semi-axes.
Given the general equation $9x^2 + 4y^2 - 36x + 24y + 36 = 0$, prove that it corresponds to an ellipse. To do so, express it in canonical form and determine the…
Given the general equation $9x^2 + 4y^2 - 36x + 24y + 36 = 0$, prove that it corresponds to an ellipse. To do so, express it in canonical form and determine the coordinates of its centre, its vertices, and its foci.
Investigating a family of ellipses with dynamic geometry
Before constructing, predict: if the foci move apart while the sum of distances stays constant, will the ellipse be more circular or more elongated?
In a dynamic geometry tool, create a slider c between 0 and 4,5, fix $a=5$ and place $F_1=(-c,0)$, $F_2=(c,0)$. Plot the equation $x^2/25+y^2/(25-c^2)=1$. Place a point P on the curve and measure its distances to the foci. As you move P, record the sum; as you change c, record the vertical semi-axis. The names of the tools depend on the program; the equation and the coordinates specify the construction.
Both curves have $a=5$; the narrower one has the foci further apart:
Present a screenshot or table and a conclusion. A classmate tries to refute it by choosing another c; discuss whether there is a counterexample or a construction error. Do not use $c=5$ in the equation: it would make a denominator zero and the locus would degenerate into a line segment. For a non-degenerate ellipse you need $2a>2c$; $c=0$ corresponds to the circular case.
Back to the start: In rooms with a whispering effect, any sound made at one focus bounces off the elliptical walls and converges exactly at the other focus, because all the bouncing paths travel the same total distance ($2a$).