Find the eccentricity of an ellipse, if its latus rectum is equal to one half of minor axis.

Find the equation to the ellipse whose focus is the point (−1, 1), whose directrix is the line x – y + 3 = 0, and whose eccentricity is .

Is the point (4, −3) within or without the ellipse 5x² + 7y² = 11?

Find the length of, and the equation to, the focal radii drawn to the point  of the ellipse 25x² + 16y² = 1600.

Prove that the sum of the squares of the reciprocals of two perpendicular diameters of an ellipse is constant.

From any point P on the ellipse  , PN is drawn perpendicular to the axis and produced to Q so that NQ equals PS, where S is a focus. Prove that locus of Q is the two straight lines y ± ex + a = 0.

A line of fixed length (a + b) moves so that its ends are always on two fixed perpendicular lines; prove that the locus of a point which divides the line into portions of length and b, is an ellipse.

Prove that the extremities of the latus rectum of the ellipse having a given major axis, a, lie on the parabola x² = – a (y – a) or on the parabola x² = a (y + a). Assume that the centre of the ellipse is at the origin and the axes coincide with the coordinate axes.

Find the equation to the tangent and normal at the point  of the ellipse 4x² + 9y² = 20.

Prove that the straight line  touches the ellipse 3x² + 4y² = 1.

Find the equation to the tangents to the ellipse 4x² + 3y² = 5 which is parallel to the straight line y = 3x + 7. Find also the coordinates of the points of contact of the tangents which are inclined at 60° to the axis of x.

Find the equation to the tangents at the end of latera recta of the ellipse  and show that they pass through the intersection of the axis and the directrix.

Find with respect to the ellipse 4x² + 7y² = 8 the polar of the point .

Find with respect to the ellipse 4x² + 7y² = 8 the pole of the line 12x + 7y + 16 = 0

Tangents are drawn from the point (3, 2) to the ellipse x² + 4y² = 9. Find the equation to their chord of contact and the equation of the straight line joining (3, 2) to the middle point of their chord of contact.

Write down the equations of the pair of tangents drawn to the ellipse 3x² + 2y² = 5 from the point (1, 2) and prove that the angle between them is

Show that the diameters whose equations are y + 3x = 0 and 4y – x = 0 are conjugate diameters of the ellipse 3x² + 4y² = 5.

A point is such that the perpendicular from the centre of the ellipse  on the polar of the point with respect to the ellipse is constant and equal to c. Show that the locus of the point is the ellipse

Prove that the locus of the pole, with respect to the ellipse  , of any tangent to the auxiliary circle is the curve

If P and D be the ends of conjugate diameters of an ellipse , find the locus of middle point of PD.

If the eccentric angle of the ends P and D, of a pair of conjugate diameters of an ellipse be ϕ and ϕ' then prove that ϕ and ϕ' differ by a right angle.

Find the locus of the point of intersection of tangents to the ellipse  which meet at right angles.

Prove that the locus of the centre of a circle which intercepts a given length 2a on the axis of X and passes through a given point on the axis of y distant b from the origin is the curve
x² – 2yb + b² = a²