The standard form of the equation of an ellipse (with its centre at origin) is
(a > 0, b > 0)
i) Point of intersection with co-ordinate axes
a) If y = 0, then x = ± a. i.e., the curve intersects X-axis at (a, 0) and (– a, 0)
(If A = (a, 0) and A' = (– a, 0), then AA' = 2a
b) If x = 0, then y = ± b. i.e., the curve intersects Y-axis at (0, b) and (0, – b).
(If B = (0, b) and B' = (0, – b), then BB' = 2b)
ii) The standard form of the equation can be rewritten as
i.e., corresponding to every real value of 'x', with | x | ≤ a, there are two values of
y – which are equal in magnitude but of opposite signs.
Similarly, corresponding to every real value of y, with | y | ≤ b, there are two real values of
x – which are equal in magnitude but of opposite signs.
Therefore, we conclude that an ellipse is symmetric about both the axes.
iii) The curve lies inside the rectangle bounded by the four lines
x = a; x = – a; y = b and y = – b. As shown in fig. (a)
iv) The part of the curve in the 1st quadrant and the complete trace is shown in
figures (b) and (c).
v) The two points (x, y) and (– x, – y) simultaneously lie on the curve. So any chord through
the centre C(0, 0) is bisected at C.