Ellipse

What is Ellipse?

An ellipse is a locus of a point which moves in such a way that its distance from a fixed point (focus) to its perpendicular distance from a fixed straight line (directrix) is constant. i.e. eccentricity(e) which is less than unity

(0 < e < 1)

Standard Equation of Ellipse

The standard equation of an ellipse is given as:

(x2 / a2) + (y2 / b2) = 1  For, a>b

So, from definition

[(SP) / (PM)] = e < 1 where P(x,y) is variable point.

(x − ae)2 + (y − 0)2 = e2 [(x – a/e) / 1]2

⇒ x2 (1 – e2) + y2 = a2 (1 – e2)

Comparing both equation

b2 = a2 (1 – e2) e2 = [(a2 – b2) / a2]

Where a ⇒ Semi major axis.

b ⇒ Semi minor axis.

Latus Rectum of an Ellipse

Chord LSL’ is called Latus rectum

(x2 / a2 + y2 / b2)= 1, Substitute (x = ae)

(a2 e2 / a2) +  (y2 / b2) = 1 y2 = b2 (1 – e2) y2 = (b4 / a2) y = ± (b2 / a) ⇒ (b2 / a), (−b2 / a)

Length of Latus Rectum =2b2a2

Note:-

  • Length of major axis = 2a
  • Length of minor axis = 2b
  • Equation of directrix, x = + a/e
  • Length of Latus Rectum = (2b2 / a)
  • Circle is a particular case of ellipse, where e = 0
  • C (0, 0) is centre of ellipse.

Position of point related to Ellipse

Let the point p(x1, y1) and ellipse

(x2 / a2) + (y2 / b2) = 1

If [(x12 / a2)+ (y12 / b2) − 1)]

= 0 {on the curve}

<0{inside the curve}

>0 {outside the curve}

Parametric form of equation of ellipse:

x = a cos Ɵ

y = b sin Ɵ

[(x2 / a2 )+ (y2 / b2)] = 1

Auxiliary circle

Some Important Equations of Ellipse

Some of the most important equations of an ellipse include tangent and tangent equation, the tangent equation in slope form, chord equation, normal equation and the equation of chord joining the points of the ellipse. All these equation are explained below in detail.

Tangent of ellipse:

The tangent of an ellipse is a line that touches a point on the curve of the ellipse. The equation and slope form of a rectangular hyperbola’s tangent is given as:

Let the equation of ellipse be [(x2 / a2) + (y2 /  b2)] = 1

the slope at point p(x1, y1)

Equation of Tangent:

y – y1 = [(− b2 x1 ) / (a2 y1)] (x – x1)

⇒ (xx1 / a2 ) + (yy1 / b2) = 1 point form

⇒at point (acosθ, bsinθ) (xcosθ / a) + (ysinθ / b) = 1

⇒ Tangent in slope form y = mx + c

Where,

Equation of Tangent in slope form:

Tangent from an external point p(x1, y1) to the ellipse

(x2 / a2) + (y2 / b2) = 1.

SS1  = T2

S ≡ (x2 / a2) + (y2 / b2) −1=0

S1 ≡ [(x12 / a2) + (y12 / b2)] – 1 = 0

T ≡ [(xx1 / a2) + (yy1 / b2)] – 1 = 0

Equation of chord with mid point (x1, y1):

The chord of an ellipse is a straight line which passes through two points on the ellipse’s curve. The chord equation of an ellipse having the midpoint as x1 and y1 will be:

T = S1

(xx1 / a2) + (yy1 / b2) = (x12 / a2) + (y12 / b2)

Equation of Normal to an Ellipse:

The normal to an ellipse bisects the angle between the lines to the foci.  The equation of the normal to an ellipse is:

Normal at point p (x1, y1)

[(x – x1) / (x1 /a2)] = [(y – y1) / (y1 / b2)]

Normal at point p (a cos Ɵ, b sin Ɵ)

The equation of chord Joining the points of Ellipse

(cosαsinα) and (cosβsinβ) can be given by:

If eccentric angles are α and β of the end of focal chords of the ellipse: