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

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 = a⁴b^4.

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.

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.

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.