Find the equation to the ellipse whose centres are the origin, whose axes are axes of coordinates and which passes through the points (1, 4) and (−6, 1)

Find the equation of the ellipse referred to centre whose minor axis is equal to the distance between the foci and whose latus rectum is 10.

Find the inclination to the major axis of the diameter of the ellipse the square of whose length is the arithmetic mean between the squares of the major and minor axis.

Find the inclination to the major axis of the diameter of the ellipse the square of whose length is the geometric mean between the squares of the major and minor axis.

Find the inclination to the major axis of the diameter of the ellipse the square of whose length is the harmonic mean between the squares of the major and minor axis.

Find the locus of the middle points of chords of an ellipse which are drawn through the positive end of the minor axis.

Q is a point on the auxiliary circle corresponding to P on the ellipse; C is the center of the ellipse. PLM is drawn parallel to CQ to meet the axes in L and M; prove that PL = b, PM = a.

Any point P of the ellipse is joined to the extremities of the major axis; prove that the portions of a directrix intercepted by them subtend a right angle at the corresponding focus.

Show that the perpendicular from the centre of an ellipse upon all chords which join the ends of perpendicular diameters, are of constant length.

If α, β, γ and δ be the eccentric angles of the four points of intersection of the ellipse and any circle, prove that α + β + γ + δ is an even multiple of π.

The tangent at any point P of a circle meets the tangent at a fixed point A in T and T is joined to B, the other end of diameter through A; prove that the locus of the intersection of AP and BT is an ellipse whose eccentricity is .

Find the equations to the tangent and normal at the point of the ellipse 5x² + 3y² = 137 whose ordinate is 2.

Find the points on the ellipse such that the tangents at each of them makes equal angle with the axis. Prove also that the length of perpendiculars from the centre on either of these tangents is .

In an ellipse referred to its centre, the length of the subtangent corresponding to the point  is , prove that the eccentricity is .

Prove that the sum of the squares of perpendiculars on any tangent to the ellipse  from two points on the minor axis, each distant  from the centre is 2a².

Find the equations to the normal at the ends of the latera recta, and prove that each passes through an end of the minor axis if e⁴ + e² = 1 (e is the eccentricity of the ellipse).

If P be a point on the ellipse  whose ordinate is y', prove that the angle between the tangent at P and the focal distance of P is

Show that the angle between the tangent to the ellipse  and the circle x² + y² = ab at their point of intersection is

A circle of radius r, is concentric with the ellipse  

 prove that the common tangent is inclined to the major axis at the angle .

The tangent at point P on the ellipse  meets the axes T and t and CY is the perpendicular on it from the centre; prove that
i) Tt . PY = a² – b² and

ii) the least value of Tt is a + b

Prove that the perpendicular from the focus of an ellipse upon any tangent and the line joining the centre to the point of contact meet on the corresponding directrix.

For the ellipse , find the tangent of the angle between CP and the normal at P and show that its greatest value is

(C is the centre and P any point on the ellipse)

Prove that the straight line lx + my = n is a normal to the ellipse , if