We invoke that an ellipse is the locus of a point which moves such that its distance from a fixed point (focus) bears a constant ratio (eccentricity) less than unity its distance from its directrix bearing a constant ratio e (0 < e < 1) .

**Ellipse**

We invoke that an ellipse is the locus of a point which moves
such that its distance from a fixed point (focus) bears a constant ratio
(eccentricity) less than unity its distance from its directrix bearing a
constant ratio *e *(0 < *e *< 1) .

Let *S *be a focus, *l *be a directrix, *e *be the
eccentricity (
0 < *e *< 1) and *P*(*x*, *y*) be the moving point.
Draw *SZ *and
*PM *perpendicular to *l
.*

Let *A *and *A*′ be the points which divide *SZ *internally
and externally in the ratio *e *:1 respectively. Let *AA*′ = 2*a *.
Let the point of intersection of the perpendicular bisector with *AA*′ be *C*.
Therefore *CA *= *a *and *CA*′ = *a*. Choose *C *as
origin and *CZ *produced as *x *-axis and the perpendicular bisector
of *AA*′ produced as *y *–axis.

By definition,

Hence we obtain the locus of *P *as which is the equation
of an **ellipse
in standard**
**form **and note that it is
symmetrical about *x *and *y *axis.

(1) The line segment *AA*¢ is called the **major axis **of the ellipse and is of length 2*a *.

(2) The line segment *BB*¢ is called the **minor axis **of the ellipse and is of length 2*b *.

(3) The line segment *CA *= the line segment *CA*¢ = **semi major axis **= ** a **and the line segment

(4) By symmetry, taking *S*¢(-*ae*, 0) as focus and *x *=- *a/e *as directrix *l*¢ gives the same ellipse.

Thus, we see that an ellipse has two foci, *S *(*ae*,
0) and *S*¢(-*ae*, 0) and
two vertices *A*(*a*, 0) and *A*¢(-*a*, 0) and
also two directrices, *x *= *a/e *and *x *=- *a/e*.

Find the length of Latus rectum of the ellipse

**Solution**

The Latus rectum *LL’ *(Fig. 5.22) of an ellipse passes through S (ae, 0) .

Hence L is (ae, y_{1}
) .

Therefore,

That is, the end points of Latus rectum L and L′ are

Hence the length of latus rectum LL' = 2b^{2} / *a*

From Fig. 5.24

The length of the major axis is 2*a *. The length of the
minor axis is 2*b *. The coordinates of the vertices are (*h *+ *a*,
*k *) and (*h *− *a*, *k *) , and the coordinates of the
foci are (*h *+ *c*, *k *) and (*h *− *c*, *k *) where *c*^{2}
= *a*^{2} − *b*^{2}.

From Fig. 5.25

The length of the major axis is 2*a* . The length of the minor axis is 2*b*. The coordinates of the vertices are * (h*, *k *+ *a*) and (h, k – a) , and
the coordinates of the foci are (h, k + c) and (h, k – c) , where c^{2}
= a^{2} - b^{2} .

The sum of the focal distances of any point on the ellipse is
equal to length of the major axis.

Let *P*(*x*, *y*) be a point on the ellipse

Draw *MM *¢ through *P * perpendicular
to directrices *l *and
*l*¢ .

Draw *PN *^ to *x *-axis.

By definition *SP *= *ePM*

= eNZ

= e[CZ - CN ]

Hence, *SP *+ *S*¢*P *= *a *- *ex *+ *a *+ *ex *= 2*a*

When b = a , the equation = 1, becomes (x - h)^{2}
+ ( y - k )^{2}
= *a*^{2} the equation of circle with centre (h, k) and radius a .

When *b *= *a*, *e *= = 0 . **Hence the eccentricity
of the circle is zero**.

Furthere, *SP/* *PM *= 0 implies *PM *→∞ . That
is, the directrix of the circle is at infinity.

Remark

**Auxiliary **circle or
circumcircle is the circle with length of major axis as diameter and **Incircle **is the circle with length of minor axis as diameter. They are given by *x*^{2} + *y*^{2} = *a*^{2} and *x*^{2} + *y*^{2} = *b*^{2} respectively.

Tags : Equation, Definition, Theorem, Proof, Example, Solution, Types , 12th Mathematics : UNIT 5 : Two Dimensional Analytical Geometry II

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12th Mathematics : UNIT 5 : Two Dimensional Analytical Geometry II : Ellipse | Equation, Definition, Theorem, Proof, Example, Solution, Types

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