Let CK is the perpendicular from the centre C of the ellipse on the polar of any point P and PM is the perpendicular from P on the same polar and is produced to meet the major axis in L. Show that i) and
ii) the product of the perpendicular from the foci on the polar
Also show that if PN is the ordinate of P and the polar meet the axis in T, show that and , where e is the eccentricity of the ellipse.
My Self Assessment
Find the locus of point of intersection of the straight lines and . Prove also that they meet at the point whose eccentric angle is 2tan−1(t).
Prove that the locus of the middle point of the portions of tangents to the ellipse included between the axes is the curve
Any ordinate NP of the ellipse meets the auxiliary circle in Q; prove that the locus of the intersection of the normals at P and Q is the circle x2 + y2 = (a + b)2.
The normal to the ellipse at P meets the axes in G and g, show that the loci of the middle points of PG and Gg are respectively the ellipses and
(e is the eccentricity of the ellipse)
Prove that the locus of the foot of the perpendicular drawn from the centre upon any tangent to the ellipse is
r2 = a2cos2θ + b2sin2θ
The normal GP of an ellipse is produced to Q so that (P is a point on the ellipse and G lies on the axis). Prove that the locus of Q is the ellipse
If the straight line y = mx + c meet the ellipse , prove that the equation to the circle described on the line joining the points of intersection as diameter is (a2 m2 + b2)(x2 + y2) + 2ma2cx – 2b2cy + c2 (a2 + b2) – a2b2 (1 + m2) = 0
In the ellipse find the equation to the chord which passes through the point (2, 1) and is bisected at that point.
In the ellipse , write down the equation to the diameter which are conjugate to the diameter whose equation is x – y = 0.
In the ellipse , write down the equation to the diameter which are conjugate to the diameter whose equation is x + y = 0
In the ellipse , write down the equation to the diameter which are conjugate to the diameter whose equation is
If the product of the perpendiculars from the force upon the polar of point P with respect to the ellipse is constant and equal to c2, prove that the locus of P is of the form b4x2 (c2 + a2e2) + c2a4y2 = a4b4.
If the pole of the normal to an ellipse at point P on it lies on the normal at another point Q on the ellipse, then show that the pole of the normal to the ellipse at Q lies on the normal at P.
Prove that the angle between the tangents that can be drawn from any point (x1, y1) to the ellipse is
If T be the point (, ) then show that the equation to the straight line joining it to foci S and S', is
Prove that the straight lines joining the centre of the ellipse to the intersection of the straight line with the ellipse are conjugate diameters.
Any tangent to the ellipse meets the director circle in p and d; prove that Cp and Cd are in the direction of conjugate diameters of the ellipse (C is the centre of the ellipse).
If CP is conjugate to normal to an ellipse at Q, prove that CQ is conjugate to normal at P (C is the centre and P and Q are points on the ellipse).
If a fixed straight line parallel to either axis meets a pair of conjugate diameters of an ellipse in the points K and L, show that the circle described on KL as diameter passes through two fixed points on the other axis.
Prove that the chord which joins the ends of a pair of conjugate diameters of an ellipse always touches a similar ellipse.
A pair of conjugate diameters of an ellipse is produced to meet the directrix; show that the orthocentre of triangle so formed is at the focus.
Tangents are drawn from any point on the ellipse to the circle x2 + y2 = r2; prove that the chord of contact are tangents to the ellipse a2x2 + b2y2 = r4
If , prove that the line joining the centre to the points of contact with the circle are conjugate diameters of the second ellipse.
