The Line x = at2 meets The Ellipse  in The Real Points Iff

The ellipse x² + 4y² = 4 is inscribed in a rectangle aligned with the coordinate axes, which in turn is inscribed in another ellipse that passes through the point (4, 0). Then, the equation of the ellipse is  

A focus of an ellipse is at the origin. The directrix is the line  x = 4 and the eccentricity is 1/2. Then the length of the semi-major axis, is  

In an ellipse, the distance between its foci is 6 and minor axis is 8. The eccentricity is 

The tangent at a point  meets the auxiliany circle in two points. The chord joining them subtends a right angle at the centre. Then, the eccentricity of the ellipse is given by  

If F₁ and F₂ be the feet of the perpendicular from the foci S₁and S₂ of an ellipse  on the tangent at any point P on the ellipse, then (S₁F₁)(S₂F₂) is equal to   

The area of the rectangle formed by the perpendiculars from the centre of the ellipse to the tangent and normal at the point-whose eccentric angle is , is  

The slope of a common tangent to the ellipse  and aconcentric circle of radius r is

P is a variable point on the ellipse  with AA’ as the major axis. Then, the maximum value of the area of the triangleAPA’ is   

The equation of the ellipse whose distance between the foci is equal to 8 and distance between the directrices is 18, is

On the ellipse 4x² + 9y² = 1, the points at which the tangents are parallel to the line 8x = 9y are