Ellipses which are drawn with the same two perpendicular lines as axes and with the sum of the reciprocals of squares of the lengths of their semi-major axis and semi-minor axis equal to a constant have only.
Four points in common
Let the two perpendicular lines be the coordinate axes and origin be the centre of the ellipse.
Let the equation of the ellipse be
It is given that (a constant). So, the equation of the ellipse becomes
This represents a family of curves passing through the intersection of
i.e., the points (±k, ±k) or, (k, k), (–k, –k), (k, –k) and (–k, k).
Hence, every member of the family passes through the four points.
The line x = at2 meets the ellipse in the real points iff
On the ellipse 4x2 + 9y2 = 1, the points at which the tangents are parallel to the line 8x = 9y are
Tangent is drawn to the ellipse , then the value of θ such that sum of intercepts on axes made by the tangent is minimum is
If p and p’ denote the lengths of the perpendicular from a focus and the centre of an ellipse with semi-major axis of length a, respectively, on a tangent to the ellipse and r denotes the focal distance of the point, then
If circumcentre of an equilateral triangle inscribed in with vertices having eccentric angle α, β, γ respectively is (x1, y1) then
Locus of the middle points of all chords of , which are at a distance of 2 units from the vertex of parabola y2 = –8axis
A point on the ellipse at a distance equal to the mean of lengths of the semi-major and semi-minor axis from the centre, is
A tangent to the ellipse is cut by the tangent at the extremities of the major axis at T and T’. The circle on TT’ as diameter passes through the point
If C is the centre and A, B are two points on the conic
4x2 + 9y2 – 8x – 36y + 4 = 0 such that ∠ACB = π/2 then CA–2 +CB–2 is equal to
The eccentricity of the ellipse with centre at the origin which meets the straight line on the axis of x and the straight line on the axis of y and whose axes lie along the axes of coordinates is