Mathematics Revision Notes of Ellipse and Hyperbola for NDA

Ellipse

An ellipse is the locus of a point which moves in a plane, so that the ratio of its distance from a fixed point (focus) and
fixed line (directrix) is constant which is less than one. For an ellipse, `e < 1`.

Let `S` be the focus, `ZM` be the directrix of the ellipse and `P (x, y)` be any point on the ellipse, then by definition, equation of ellipse is

`x^2/a^2 +y^2/b^2=1`

where `b^2= a^2 (1-e^2)`

since, `e < 1`

`=> a^2(1-e^2) < a^2`

`=> b^2 < a^2`

The other form of equation of ellipse is `x^2/a^2 +b^2/b^2 =1`

where, `a^2 = b^2 (1- e^2)` i.e, `a < b`

Definition and Basic Terminology of an Ellipse

(i) Focal axis Line containing two fixed points `F_1` and `F_2`, is called focal axis or major axis.

(ii) Focal length Distance between `F_1` and `F_2`, is called focal length or focal distance, i.e. `F_1 F_2 = 2ae`.

(iii) Diameter Any chord of the ellipse passing through its centre, is called diameter.

(iv) Principal axis Major and minor axes together, known as principal axis.

(v) Latusrectum Line perpendicular to major axis and passesing through focus, is called latusrectum.

Important Terms of an Ellipse

There are two standard forms of ellipse with centre at the origin and axes along coordinate axes. The foci of the ellipse are either on the `x`-axis or on they-axis.

1. Major Axis Along `x`-axis :
The equation of this type of ellipse is of the form `x^2/a^2+y^2/b^2=1` , where `a>b>0` and `b =a sqrt(1-e^2)` .

For this ellipse :

(i) Major axis is `2a`

(ii) Minor axis is `2b`.

(iii) Centre is `(0, 0)`

(iv) Vertices are `(pm a, 0)`

(v) Foci are `(pm a e, 0)`

(vi) Equation of directrices are `x= pm a/e`

(vii) Equation of major axis is `y =0`

(viii) Equation of minor axis is `x = 0`

(ix) Length of latus rectum `= (2b^2)/a`

(x) Extremity of latus rectum is `(+-a e,+-b^2/a)`

2. Major Axis Along `y`-axis :
The equation of this type of ellipse is of the form `x^2/a^2+y^2/b^2=1`. where `0 < a < b` and `a=bsqrt(1-e^2)`

For this ellipse:

(i) Major axis is `2b`

(ii) Minor axis is `2a`.

(iii) Centre is `(0, 0)`

(iv) Vertices are `(0, pm b)`

(v) Foci are `(0, pm be)`

(vi) Equation of directrices are `y = pm b/e`

(vii) Equation of major axis is `x = 0`

(viii) Equarion of minor axis is `y =0`

(ix) Length of latus rectum `=(2a^2)/b`

(x) Extremity of latus rectum is `( pm a^2/b, pm be)`

Parametric Form of an Ellipse

Let the equation of ellipse in standard form be `x^2/a^2 +y^2/b^2 =1`

Then, the equations of an ellipse in the parametric form will be

`x =a cos phi, y = b sin phi`, where `phi` is eccentric angle.

Position of a Point w.r.t. Ellipse

Let standard equation of an ellipse be `E : x^2/a^2 +y^2/b^2 -1=0` then the point `P (x_1 , y_1 )` will lie outside the ellipse, if `E (P) > 0` lie
inside, if `E(P) < 0` and en the ellipse, if `E (P) = 0`.

Condition for Intersection of a Line and an Ellipse

Let us consider a line `y = mx + c` and an ellipse `x^2/a^2 +y^2/b^2 =1` then the line will touch,
intersect and do not intersect according as `a^2m^2 + b^2 = c^2, a^2m^2 + b^2 > c^2` and `a^2m^ 2 + b^2 < c^2`, respectively

Hyperbola

A hyperbola is the locus of a point which moves in a plane such that the ratio of its distance from a fixed point and a given straight line is always constant.

The fixed point is called the focus, the fixed line is called the directrix and the constant ratio is called the eccentricity of the hyperbola and denoted by `e.`

In the given figure, Sis the focus and N'N the directrix.

Let P be any point on the hyperbola, then `(OS)/(PM) = e, e > 1`

Equation of a hyperbola can be obtained if the coordinates of its focus, equation of its directrix and eccentricity are given.



STANDARD EQUATION OF A HYPERBOLA :

Let S be the focus & ZN is the directrix of an ellipse. Draw perpendicular from S to the directrix which meet it at Z. A moving point is on the hyperbola such that

`PS = ePM`

`(PS)^2 = e^2 (PM)^2`

`=> (x-ae)^2 +(y-0)^2 =e^2(x- a/e)^2`

`x^2/a^2 -y^2/(a^2(e^2-1))=1`

Hence equation of hyperbola is `x^2/a^2 -y^2/b^2 =1` where `b^2 = a^2 ( e^2 - 1)`

Terms Related to Hyperbola

(1) Centre :
In the figure, `C` is the centre of the ellipse. All chords passing through `C` are called diameter and bisected at `C`.

(2) Foci :
`S(a e, 0)` and `S'(- a e, 0)` are two foci of hyperbola. Line containing the fixed points `S` and `S'` (called Foci) is called Transverse Axis `(TA)` or Focal Axis and the distance between `S` and `S'` is called Focal Length.

(3) Axes :
The line `A A'` is called transverse axis and the line `B B'` is perpendicular to it and passes through the centre `(0, 0)` of the hyperbola is called conjugate axis.

The length of transverse and conjugate axes are taken as `2a` and `2b` respectively.

The transverse and conjugate axes together are called principal axes of hyperbola and their intersection point is called the centre of hyperbola.

The points of intersection of the directrix with the transverse axis are known as Foot of the directrix (`Z` and `Z'` ).

(4) Vertex :
The points of intersection `(A, A')` of the curve with the transverse axis are called Vertices of the hyperbola.

(5) Double Ordiante :
Any chord perpendicular to the Transverse axis is called a Double Ordinate.

(6) Latus-Rectum :
When double ordinates passes through the focus of parabola then it is called the latus rectum. In the given figure `L L'` and `M M'` are the latus-rectums of the hyperbola.

let `L L' = 2k` then `LS = L'S = k`

Let `L(ae, k)` lie on the hyperbola `x^2/a^2-y^2/b^2=1`

`:. (a^2e^2)/a^2-k^2/b^2=1`

or `k^2=b^2(e^2-1)=b^2(b^2/a^2)` `[:. b^2=a^2(e^2-1)]`

`:. k=b^2/a`

`:. 2k=(2b^2)/a=L L'`

= `(2e)` (distance between the focus and the foot of the corresponding directrix)

End points oflatus -rectums are

`L=(ae,b^2/a), L'=(ae,-b^2/a); M=(-ae,b^2/a); M'=(-ae,-b^2/a)` respectively

(7) Focal Chord :
A chord of hyperbola passing through its focus is called a focal chord.

(8) Eccentricity :
For the hyperbola `x^2/a^2-y^2/b^2=1` we have

`b^2=a^3(e^2-1)`

`=> e= sqrt(1+(b^2/a^2))=> e= sqrt({1+(text(conjugate axis)^2)/(text(transverse axis)^2)})`

Eccentricity defines the curvature of the hyperbola and is mathematically spelled as:

`e = (text(distance from centre to focus))/(text(distance from centre to vertex))`

(9) Focal Distance of a Point On Hyperbola :
The hyperbola is `x^2/a^2-y^2/b^2=1` ...................(1)

The foci `S` and `S'` are `(ae, 0)` and `(- ae, 0)` & corresponded directrix are `x =a/e` and `x = -a/e` respectively.

Let `P(x_1, y_1)` be any point on `(1)`.

Now `SP = ePM = e(x_1-a/e)=ex_1-a`

and `S'P = ePM' = e(x_1+a/e)=ex_1+a`

`:. S'P - SP = (ex_1 + a) - (ex_1 - a) = 2a`

`= A A' =` Transverse axis

Thus hyperbola is the locus of a point which moves in a plane such that the difference of its distances from two
fixed point (foci) is constant and always equal to transverse axis.

Hence, in the given figure

`PS' - PS = QS' - QS = RS - RS' =` length of transverse axis.


Conjugate Hyperbola

The hyperbola whose transverse and conjugate axes are respectively the conjugate and
transverse axes of a given hyperbola, is called conjugate hyperbola of given hyperbola.

Equation of conjugate hyperbola is ` - x^2/a^2 +y^2/b^2=1`

Important Terms of Hyperbola

Parametric Equations of Hyperbola

The equations `x =a sec theta` and `y = b tan theta` are known as the parametric equations of the hyperbola `x^2/a^2-y^2/b^2= 1`. The point `(a sec theta, b tan theta)` will lie on the hyperbola for all values of `theta`

Condition for Intersection of Line and Hyperbola

Let us consider a line `y = mx + c` and a hyperbola `x^2/a^2 -y^2/b^2 =1`

By above two equations, we have `x^2/a^2 -((mx+c)^2)/b^2 =1`

This line will intersect, touch or do not intersect the hyperbola according to discriminant of above equation, i.e.

(i) If `D > 0`, then line will intersect the hyperbola at two points.

(ii) If `D = 0`, then `c^2 = a^2m^2 - b^2` touches the parabola.

(iii) If `D < 0`, then line is outside the hyperbola.

Position of a Point w.r.t. Hyperbola

Any point `P(x, y)` lie on the hyperbola

`H : x^2/a^2 -y^2/b^2 -1=0`

According to the following conditions:

(i) Outside the hyperbola, `H(P) < 0`

(ii) Inside the hyperbola, `H(P) > 0`

(iii) On the hyperbola, `H(P) = 0`

Rectangular or Equilateral Hyperbola

A hyperbola for which `a = b` is said to be rectangular or equilateral. Its equation is `x^ 2- y^2= a^2`
.

Conjugate Hyperbola

`y^2/b^2 -x^2/a^2 =1` is known as conjugate hyperbola to the hyperbola `x^2/a^2 -y^2/b^2 =1` If `e_1` and `e_2` are their eccentricities, then

`e_1^2 = 1+ b^2/a^2`

and `e_2^2 =1+ a^2/b^2`

On solving Eqs. (i) and (ii), we get

`1/(e_1^2)+ 1/(e_2^2) =1`

Some Important Results of Hyperbola

(i) Differenqe of the focal distances, i.e. `PS'- PS =` constant `=> 2a =` Length of transverse axis

(ii) Eccentricity of the rectangular hyperbola `= sqrt 2` and angle between asymptotes `= 90^o`.

 
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