The Locus Of The Foot Of The Perpendicular From The Foci On Any Tangent To The Ellipse 

Tangents are drawn to the ellipse  and the circle x² + y² = a² at the points where a common ordinate cuts them (on the same side of the x-axis). Then, the greatest acute angle between these tangents is given by 

The area of the quadrilateral formed by the tangents at the end-points of latusrecta to the ellipse 

If α – β = constant, then the locus of the point of intersection of tangents at  to the ellipse   

Let S(3, 4) and S’(9, 12) be two foci of an ellipse. If the coordinates of the foot of the perpendicular from focus S to a tangent to the ellipse is (1, –4), then the eccentricity of the ellipse is 

Let S and S’ be two foci of the ellipse . If a circle described on SS’ as diameter intersects the ellipse in real and distinct points, then the eccentricity e of the ellipse satisfies   

If PSQ is a focal chord of the ellipse , then the harmonic mean of SP and SQ is  

If PSQ is a focal chord of the ellipse 16x² + 25y² = 400 such that SP = 8, then SQ =

If S and S’ are two focii of the ellipse 16x² + 25y² = 400 andPSQ is a focal chord such that SP = 16, then S’Q = 

Tangent at a point on the ellipse  is drawn which cuts the coordinates axes at A and B. The minimum area of the triangleOAB is (O being origin)

The locus of the point of intersection of tangents to the ellipse  at the points whose eccentric angles differ by  is