1 Standard Ellipse (e < 1)
| Ellipse Imp.terms | {(x2/a2) + (y2/b2) = 1} |
| Centre | (0,0) | (0,0) |
| Vertices | (±a,0) | (0, ±b) |
| Length of major axis | 2a | 2b |
| Length of minor axis | 2b | 2a |
| Foci | ( ±ae,0) | (0, ± be |
| Equation of directories | x = ± a/e | y = ± b/e |
| Relation in a,b and e | b2 = a2(1-e2) | a2 = b2 (1-e2) |
| Length of rectum | 2b2/a | 2a2/b |
| Ends of lotus rectum | { ± ae, ± (b2/a)} | { ± (a2/b) , ± be } |
| Parametric Coordinates | (a cos Æ , b sin Æ ) | ( a cos Æ , b sin Æ
0 £ Æ < 2 p |
| Focal radii | SP= a – ex1
S’P = a + ex1 | SP = b – ey1
S’P = b + ey1 |
| Sum of focal radii | SP + S’P | 2a 2b |
| Distance btn foci | 2ae | 2be |
| Distance btn directories | 2a/e | 2b/e |
| Tangents at the vertices | x = -a , x = a | y = b, y = - b |
Note:
If P is any point on ellipse and length of perpendiculares form to minor axis and major axis are P1 & P2 then |xp| = P1, |yp|= P2
2 Special form of ellipse:
If the centre of an ellipse is at point (h,k) and the direction of the axes are parallel to the coordinate axes, then its is 
3 Auxiliary Circle:
The circle described by taking centre of an ellipse as centre and major axis as a diameter is called an auxiliary circle of the ellipse. If (x2/a2) + (y2/b2) = 1 is an ellipse then its auxiliary circle is x2 + y2 = a2 .
Note:
Ellipse is locus of a point which moves in such a way that it divides the normal of a point on diameter of a point of circle in fixed ratio.
4 Position of a point and a line w.r.t . an ellipse :
* The point lies outside, on or inside the ellipse if
S1 = (x21 / a2) + (y21 / b2) – 1 > , = or < 0
* The line y = mx + c does not intersect, touches, intersect the ellipse if a2m2 + b2 < = > c2
5 Equation of tangent in different forms:
(i) ellipse (x2 / a2) + (y2 / b2) = 1 at the point (x1, y1) (xx1 / a2) + (yy1 / b2) = 1
(ii) Slope form: if the line y = mx + c touches the ellipse (x2/a2 ) + (y2/b2) = 1 then c2 = a2m2 + b2 . Hence the straight line y = mx ± Ö(a2m2+ b2) always represents the tangents to the ellipse.
Point of contact:
Line y = mx Ö(a2m2 + b2) touches the ellipse
(iii) Parametric form: the equation of tangent at any point ( acos Æ , b sin Æ) is (x/a) cos Æ + (y/b) sin Æ = 1
6 Equation of pair of tangents :Equation of pair of tangents form (x1, y1 ) to an ellipse (x2 / a2 ) + (y2/ b2) = 1 is given by SS1 = T2
7 Equation of normal in different forms:
(i) Point form: The equation of the normal at (x1, y1) to the ellipse (x2/a2 ) + (y2/b2) = 1 is (a2x / x1) - (b2x / y1) = a2 - b2.
(ii) Parametric form: the equation of the normal to the ellipse (x2/a2) + (y2/b2) = 1 at (a cos Æ , b sin Æ ) is ax sec Æ - by cosec Æ = a2 - vb2.
(iii) Slope form: If m is the slope of the normal to the ellipse (x2/a2) + (y2/b2) = 1 then the equation of normal
The co-ordinates of the point of contact are
Note:
In general three normal’s can be drawn form a point (x1, y1) to an ellipse (x2/a2) + ( y2/b2) = 1
8 Properties of tangents & normal’s :
(i) product of length of perpendicular form either foci to any tangent to the ellipse will be equal to square of semi minor axis.
(ii) The locus of foot of perpendicular drawn from either foci to any tangent lies on auxiliary circle
(iii) The circle drawn on any focal radius as diameter will touch auxiliary circle.
(iv) The protion of the tangent intercepted between the point and directrix makes right angle at corrsponding focus.
(v) Sum of square of intercept made by auxiliary circle on any two perpendicular tangents of an ellipse will be constant.
(vi) If a light ray originates from one of foci , then it will pass through the other focus after reflection from eilipse .
9 Equation of chord of contact :Equation of chord of contact of the tangents drawn form the external point (x1, y1) to an eilipse is given by (xx1/a2) + yy1 / b2) = 0 i.e. T =0
10 The equation of a chord of an ellipse :The equation of a chord of an ellipse (x2/a2) + (y2/b2) = 1 whose mid point is (x1, y1) is T = S1.
11 Equation of chord joining the points :Equation of chord joining the points ( a cos q, b sin q) and
(acos Æ, b sin Æ) on the ellipse (x2 / a2) + (y2 / b2) =1 is (x/a)cos (q+Æ / 2) + (y/b) sin (q+Æ /2) = cos (q-Æ /2)
(i) Relation between eccentric angles of focal chord
Þ tan (q1 /2), tan (q2 /2) = (±e-1)/(1±e)
(ii) Sum of feet of eccentric angles is odd p.
i.e. q1 + q2 + q3 + q4 = (2n + 1) p
12 Equation of polar of the point :Equation of polar of the point (x1, y1) w.r.t. the ellipse (x2 / a2) + y2 / b2) =1 is given by (xx1 a2 ) + (yy1/b2) = 0 i.e. T =0
The pole of the line ex + my + n = 0 w.r.t. the ellipse
13 Eccentric angles of the extremities :Eccentric angles of the extremities of latus rectum of the ellipse (x2 / a2) + ( y2 / b2) =1 are tan-1 (±(b/ae).
14 (i) Equation angle of the diameter bisecting the chords of slope in the ellipse (x2 / a2) + (y2 /b2) = 1 is y = - (b2 / a2m)x
(ii) Conjugate Diameters : The straight lines y = m1x, y = m2x are conjugate diameters of the ellipse (x2 /a2) + (y2/b2) =1 if m1m2 = - (b2/a2)
(iii) Properties of conjugate diameters:
(a) If CP and CQ be two conjugate semi-diameters of the ellipse (x2/a2) + (y2/b2) = 1 then CP2 + CQ2 = a2 + b2
(b) If q and Æ are the eccentric angles of the extremities of two conjugate diameters, then q-Æ = ± (p/2)
(c) If CP, CQ be two conjugate semi-diameters of the ellipse (x2/a2) + ( y2 / b2) =1 and S, S’ be two foci of the ellipse, then SP.S’P = CQ2
(d) The tangents at the ends of a pair of conjugate diameters of an ellipse form a parallelogram.
15 The area of the parallelogram formed:The area of the parallelogram formed by the tangents at the ends of conjugate diameters of an ellipse is constant and is to the product of the axis i.e. 4ab.
16 Length of subtangent and subnormal:Length of subtangent and subnormal at P(x1, y1) to the ellipse
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