If circles be described on two semi – conjugate diameters of the ellipse  as diameters, prove that the locus of their second points of intersection of the circles is the curve

2(x² + y²)² = a²x² +b²y²

The tangents drawn from a point P to the ellipse  make angles θ₁ and θ₂ with the major axis. Find the locus of P when θ₁ + θ₂ is a constant (=2α)

The tangents drawn from a point P to the ellipse  make angles θ₁ and θ₂ with the major axis. Find the locus of P when tanθ₁ + tanθ₂ is a constant (=c)

The tangents drawn from a point P to the ellipse  make angles θ₁ and θ₂ with the major axis. Find the locus of P when tanθ₁ − tanθ₂ is a constant (=d)

Tangents are drawn from a point P to the ellipse  make angles θ₁ and θ₂ with the major axis. Find the locus of P when tan²θ₁ – tan²θ₂ is a constant (=λ)

Find the locus of the intersection of the tangents to the ellipse  which meet at a given angle α.

Find the locus of the intersection of the tangents to the ellipse  if the sum of the eccentric angles of their points of contact be equal to a constant angle 2α.

Find the locus of the intersection of the tangents to the ellipse  if the difference of their eccentric angles of the points of contact is 120°.

Find the locus of the intersection of the tangents to the ellipse  if the line joining the points of contact to the centre be perpendicular.

Find the locus of the intersection of the tangents to the ellipse  if the sum of ordinates of the point of contact be equal to b.

Find the locus of the middle points of the chord of the ellipse  whose distance from the centre is the constant length c.

Find the locus of the middle points of the chord of the ellipse  which subtend a right angle at the centre.

Find the locus of the middle points of the chord of the ellipse  which passes through the given point (h, k)

Find the locus of the middle points of the chord of the ellipse  whose length is constant (= 2c).

Find the locus of the middle points of the chord of the ellipse  whose poles are on the auxiliary circle.

Find the locus of the middle points of the chord of the ellipse  tangents at the end of which are at right angles.

A parallelogram circumscribes the ellipse and two of its opposite angular points lie on the straight lines x² = h²; prove that locus of the other two is the conic

Circles of constant radius c are drawn to pass through the ends of a variable diameter of the ellipse. Prove that the locus of their centres is the curve
(x² + y²) (a²x² + b²y² + a²b²) = c² (a²x² + b²y²)

The polar of a point P with respect to an ellipse  touches a fixed circle whose centre is on the major axis and which passes through the centre of the ellipse. Show that the locus of P is a parabola whose latus rectum is equal to the product of the square of the latus rectum of the ellipse and the reciprocal of the diameter of the circle.

Show that the locus of the pole, with respect to auxiliary circle, of a tangent to the ellipse , is a similar concentric ellipse whose major axis is at right angles to that of original ellipse.

Chords of the ellipse touch the parabola ay² = −2b²x ; prove that the locus of their poles is the parabola ay² = 2b²x

The ellipse is rotated through a right angle in its own plane about its centre, which is fixed; prove that the locus of the point of intersection of a tangent to the ellipse in the original position with the tangent at the same point of the curve in the new position is
(x² + y²) (x² + y² – a² – b²) = 2(a² – b²)xy

Find the locus of the intersection of tangents at the ends of chords of the ellipse , which are of constant length 2c.