ELLIPSE

Definition :     

    The ellipse is the intersection of double napped cone with an oblique plane.

An ellipse is the locus of a point in a plane which moves so that its distance from a fixed point bears a constant ratio e (0 < e < 1) to its distance from a fixed line. The fixed point is called the focus S and the fixed line is callled the directrix d .

If S is a fixed point is called focus and directrix d is a fixed line not containing the focus then by definition PS/PM = e and PS = e PM  , where PM is the perpendicular on the directrix and e is the real number with 0 < e < 1 called eccentricity of the ellipse.

Standard Equation of Ellipse :

     Let's derive the standard equation of the ellipse 

Let S be the focus , d be the directrix and e be the eccentricity of an ellipse.

Let  SZ perpendicular to directrix d . A ans A ' divide the segment SZ internaly as well as externally in the ratio e : 1 .

Let AA ' = 2a , midpoint O of segment AA ' be the origine . Then O ≡ (0, 0) , A ≡  (a, 0) and A ' ≡ (-a, 0) 

By definition of ellipse A and A ' lie on ellipse 

 SA/AZ = e/ 1      and   SA '/ A ' Z = -e/1

Let P(x, y) be any point on ellilse .

Since P is on the ellipse SP = e PM...(1)
therefor SA = e AZ.

Let  Z ≡ (k, 0)  and S ≡ (h, 0)

By section formula 

a e + a = e k + h   ....(2)

-a e + a = e k - h .....(3)

Solving these equation , we get 

k = a/e      and h = a e 

Focus  S ≡ (ae, 0)  and  Z ≡  (a/e, 0)

Equation of the directrix is x = a/e 

This is x - a/e = 0

SP  =  Focal distance 

PM = distance of point P from directrix 

From equation (1), (4) and (5)

Squaring both side .

(x - ae)² + (y - 0)² = e²x²  - 2 aex  + a²

  x²  - 2 aex + a²e² + y² = e²x² - 2 aex + a²

x² + a²e² + y² = e²x² + a² 

(1 - e²)x² + y²  = a² (1 - e²)

Sience (1 - e²)  >  0 , Dividing both side by a² (1 - e²)

This is the standard equation of ellipse .

Note :

i)  The ellipse intersects x- axis A (a, 0) , A' (-a, 0) and y-axis B(0, b) , B' (0. -b) , these are the vertices of the ellipse.

ii)  The line segment through the foci of the ellipse is called the major axiz and the line segment through centre and perpendicular to major axix is the minor axiz . The major axiz and minor axis together are called principlap axis of the ellipse. 

iii)   The segment AA ' of length 2a is called the major axis and th segment BB' of the length 2b is called the minor axis . Ellipse is symmeetric about both the axes.

iv)  Latus rectum is the chord through focus which is perpendicular to major axis . It is bisected at the focus . These are two latera recta as there are two foci.

Tangent to an ellipse :

   A straight line which intersect the curve ellipse in two coincident point is called a tangent to the ellipse .

To find the equation of tangent to the elllipse x² /a²  + y² /b² = 1  at the point  P (x_{1 ,} y₁)  on it . Hence to obtain the equation of tangent at P(θ₁).

We need to know the slope of the tangent at  P (x_{1 ,} y₁) . From the theory of derivative of a function , the slope of the tangent is dy/dx at  (x_{1 ,} y₁)

The equation of ellipse is  x² /a²  + y² /b² = 1  differentiate both sides with respect to x 

∴Equation of the tangent (by slope point form)

at   P (x_{1 ,} y₁) is y - y₁= - (b² /a²)(x₁- y₁)(x -x₁ )

 a²  y₁(y- y₁) = - b²x₁(x - x₁)

 b²  y₁y  - a² y₁ = -  b² x x + b x 

 b² x₁ x + a² y₁y  =  b² x₁²  +  a² y₁ ² 

Dividing by 

Now  P (x_{1 ,} y₁) lies on the ellipse 

Thses is the equation of the tangent at P (x_{1 ,} y₁) on it 

Now θ₁ is the parameter of point P

∴  P (x_{1 ,} y₁) = (a cosθ₁  , b sinθ₁)  that is 

x₁ = a cosθ₁    , y₁  = b sin θ₁ 

Substituting these values in equation (1) 

is the required equation of the tangent at P (θ₁) .