Definition: Locus of point which moves in a plane such that its distance from a fixed point is in a constant ratio to its distance from a fixed line where constant ratio e < 1 is called ellipse.
Standard equation of ellipse is (a > b)
1) Centre (0, 0).
2) Foci (ae, 0), (-ae, 0).
3) Directrices x =
4) Major axis is y = 0 (i.e. x-axis)
Minor axis is x = 0 (i.e y-axis)
Double ordinate: If P be a point on the Ellipse drawn PN perpendicular to the axis of the ellipse and produced to meet the curve again at P' then PP' is called a doulde ordinate.
Latus Rectum: A double ordinate passing through the focus is called Latus rectum.
length of Latus rectum =
End points of latus rectum are
Focal chord: A chord of the ellipse passing through its focus is called a focal chord.
Vertices: The vertices of the ellipse are the points of intersection of ellipse and major axis
A(a, 0), A'(-a, 0).
Let be ellipse. Then the circle whose extremities of diametre are vertices of ellipse is called Auxiliary circle of ellipse.
Auxiliary circle of ellipse is x2 + y2 = a2
Parametric equation of ellipse:
Let P be a point on the ellipse. Ordinate PN produced to meet the auxiliary circle at Q. Line OQ makes an angle θ with +ve direction of axis.
Then coordinates of Q (a cosθ, a sinθ).
coordinates of P(a cosθ, b sinθ)
x = a cos θ, y = b sinθ are called parametric equations of the ellipse.
Let P be any point on the ellipse and S, S' are foci. SP, S'P are called focal distances.
P(x1, y1) be any point on the ellipse and S(ae, 0), S'(-ae, 0) are foci,
then SP = a - ex1, S'P = a + ex1
Sum of focal distances of any point on the ellipse is equal to major axis 2a.
(b > a)
P(x1, y1), Q(x2, y2) are two points on the ellipse. Then equation of chord passing through
P(x1, y1) Q(x2, y2) is S1 + S2 = S12
P(a cosα, b sinα), Q(a cosβ, b sinβ). Equation of chord passing through PQ is
If the above chord passing through focus (ae, 0) then
If it is passing through (-ae, 0) then
The point P(x1, y1) lies outside, on, or inside the ellipse according as S11 > = < 0.
Equation of tangent at (a cosθ, b sinθ) to ellipse
If y= mx + c is tangent to the ellipse then c2 = a2m2 + b2
For any value of m, is tangent to
Equation of normal at P(x1, y1) to the ellipse,
Equation of normal to the ellipse at (a cosθ, b sinθ) is
Equations of the normals of slope 'm' to the ellipse
If the Normal at one end of latus rectum of an ellipse passes through one extremity of the minor axis, then e4 + e2 - 1 = 0
If the straight line lx + my + n = 0, is normal to the ellipse
In general four normals can be drawn to an ellipse from a point and If α, β, γ, δ are eccentric angles of these four conormal points then α + β + γ + δ = (2n + 1)π, n Z
If the normals at P, Q, R, S to the ellipse intersects at T(h, k) then the points lies on the curve (a2 - b2)xy + b2kx - a2hy = 0
This curve is called apollonian rectangular hyperbola.
The combined equation of the pair of tangents drawn from a point (x1, y1) to the ellipse
The equation of chord of contact of tangents drawn from a point P(x1, y1) to the ellipse
The locus of point of intersection of perpendicular tangents to the ellipse is circle called director circle, Equation is x2 + y2 = a2 + b2.
The Equation of a chord of the ellipse
bisected at the point P(x1, y1)is
Locus of point intersection of tangents drawn from the extremity of the chord passing through the fixed point is a straight line is called polar of that point, and point is called pole of that line with respect to ellipse.
If P(x1, y1) is a point then polar of P(x1, y1) with respect to Ellipse is S1 = 0
Pole of the line lx + my + n = 0 with respect to ellipse
Polar of the focus is directrix.
If the polar of P(x1, y1) passes through Q(x2, y2) then polar of Q(x2, y2) passes through
P(x1, y1) and such points are called conjugate points.
If the pole of the line l1x +m1y + n1 = 0 lies on l2x + m2y + n2 = 0 then pole of
l2x + m2y + n2 = 0 lies on l1x + m1y + n1 = 0 such lines are called conjugate lines.
If P(x1, y1), Q(x2, y2) are conjugate points then S12 = 0
If l1x + m1y + n1 = 0, l2x + m2y + n2 = 0 are conjugate lines then a2l1l2 + b2m1m2 = n1n2
The locus of middle points of a system of parallel chords of an ellipse is called a diameter.
Equation of diameters whose slope of parallel chords is m is
Two diameters are said to be conjugate when each bisect all chords parallel to the other.
If y = m1x, y = m2x are two conjugate diameters then
Co-ordinates of the four extremities of two conjugate diameters are P(a cosθ, b sinθ), P' (-a cosθ, -b sinθ), Q(-a sinθ, b cosθ), Q'(a sinθ, - b cosθ)
The eccentric angles of the ends of a pair of conjugate diameters of an ellipse differ by a right angle.
The sum of the squares of any two conjugate semi diameters an ellipse is constant and equal to sum of squares of the semi axes of the ellipse.
CP2 + CQ2 = a2 + b2
The product of the focal distances of a point on an ellipse is equal to the square of the semi diameter which is conjugate to the diameter through the point.
SP.S'P = CQ2
The tangents at the extremities of a pair conjugate diameters form a parallelogram whose are is constant and equal to the product of the axes (4ab)
If the normal at P meets the axis of the ellipse at G then SG = eSP, S'G = eS'P
... normal is internal bisector of S'PS
... tangent is external biscetor of S'PS
If an incoming light ray passes through one focus (s) strike the concave side of the ellipse then it will get reflected towards other focus (S')
The Position of the tangent between the curve and directrix subtends right angle at the corresponding focus.
The line segment of the tangent between the tangents at the vertices subtends right angle at the foci.
Foot of the perpendicular from foci to any tangent lies Auxiliary circle, and product of perpendiculars is b2 (square of semi minor axis).
If the normal at any point P on the ellipse meets the axes in G and g, respectively then find the ratio PG : pg = b2 : a2