# 4.5 Ellipse (0 < *e* < 1)

An *ellipse* with eccentricity *e* (where 0 < *e* < 1) is the set of points *P* in the plane whose distances from a fixed point *F* are *e* times their distances from a fixed line *d*. We obtain such an ellipse *in standard form* if

the focus

*F*lies on the*x*-axis, and has coordinates (*ae*, 0), where*a*> 0;the directrix

*d*is the line with equation*x*=*a*/*e*.

Let *P* (*x*, *y*) be an arbitrary point on the ellipse, and let *M* be the foot of the perpendicular from *P* to the directrix. Since *FP* = *e* × *PM*, by the definition of the ellipse, it follows that *FP*^{2} = *e*^{2} × *PM*^{2}; that is,

Multiplying out the brackets, we obtain

*x*^{2} − 2*aex* + *a*_{2}*e*_{2} + *y*_{2} = *e*_{2}*x*_{2} − 2*aex* + *a*_{2},

which simplifies to the equation

*x*_{2}(1 − *e*_{2}) + *y*_{2} = *a*_{2}(1 − *e*_{2}),

that is,

Substituting *b* for , so that *b*^{2} = *a*^{2}(1 − *e*^{2}), we obtain the standard form of the equation of the ellipse

This equation is symmetric in *x* and in *y*, so that the ellipse also has a second focus *F*′ at (−*ae*, 0), and a second directrix *d*′ with equation *x* = −*a*/*e*.

The ellipse intersects the axes at the points (±*a*, 0) and (0, ±*b*). We call the line segment joining the points (±*a*, 0) the *major axis* of the ellipse, and the line segment joining the points (0, ±*b*) the *minor axis* of the ellipse. Since *b* < *a*, the minor axis is shorter than the major axis. The origin is the centre of this ellipse.

*Note:* Since 0 < *e* < 1, we have 0 < *b* < *a*.

Each point with coordinates (*a* cos *t*, *b* sin *t*) lies on the ellipse, since

Then, just as for the parabola, we can check that

gives a parametric representation of the ellipse.

An ellipse with eccentricity *e* = 0 is a circle. In this case, *a* = *b* and the circle *x*^{2} + *y*^{2} = *a*^{2} can be parametrised by

We summarise these facts about ellipses (including circles) as follows:

## Ellipse in standard form

An ellipse in standard form has equation

It can also be described by the parametric equations

If *e* > 0, it has foci (±*ae*, 0) and directrices *x* = ±*a*/*e*; its major axis is the line segment joining the points (±*a*, 0), and its minor axis is the line segment joining the points (0, ±*b*).

If *e* = 0, the ellipse is a circle.

## Example 56

Let *P* be a point , *t* ∈ , on the ellipse with equation *x*^{2} + 2*y*^{2} = 1.

**(a)**Determine the foci*F*and*F*′ of the ellipse.**(b)**Determine the gradients of*FP*and*F*′*P*, when these lines are not parallel to the*y*-axis.**(c)**Find those points*P*on the ellipse for which*FP*is perpendicular to*F*′*P*.

### Answer

**(a)**This ellipse is of the form with*a*= 1 and , so If*e*denotes the eccentricity of the ellipse, so that*b*^{2}=*a*^{2}(1 −*e*^{2}), then we haveit follows that , so .

In the general case, the foci are (±

*ae*, 0); it follows that here the foci are .**(b)**Let*F*and*F*′ be and , respectively. (It does not matter which way round these are chosen.)Then the gradient of

*FP*iswhere we know that , since

*FP*is not parallel to the*y*-axis.Similarly, the gradient of

*F*′*P*iswhere we know that , since

*F*′*P*is not parallel to the*y*-axis.**(c)**When*FP*is perpendicular to*F*′*P*, we haveWe may rewrite this in the form

so 2 cos

^{2}*t*− 1 + sin^{2}*t*= 0.Since cos

^{2}*t*+ sin^{2}*t*= 1, it follows that cos^{2}*t*= 0 and so cos*t*= 0. This occurs only when*t*= ±/2; that is, at the points .