The Locus Of The Point Of Intersection Of Tangents To An Ellipse At Two Points, Sum Of Whose Eccentric Angles Is Constant Is A/an

Find the locus of the centroid of an equilateral triangle inscribed in the ellipse     

If SY and S₁Y₁ be perpendiculars from the foci upon the tangent atP of an ellipse, then Y and Y₁ lie on the auxiliary circle andSY.S₁Y₁ =    

Find the condition on a and b for which two distinct chords of the ellipse  passing through (a, –b) are bisected by the line x + y = b. 

Let P be a point on the ellipse , 0 < b < a. Let the line parallel to y-axis passes through P meet the circle  at the point Q, such that P and Q are the same side of x-axis. For two positive  real numbers r and s, find the locus of the point Ron PQ such that PR : RQ = r : s as P varies over the ellipse.  

Consider the family of circle x² + y² = r², 2 < r < 5. If in the first quadrant the common tangent to a circle of the family and the ellipse 4x² + 25y² = 100 meets the coordinate axes at A and B, then find the equation of the locus of the mid point of AB.

The orbit of earth is an ellipse with eccentricity 1/60 with the sun at one focus the major axis being approximately 186 × 10⁶ miles in length. Find the shortest and longest distance of the earth from the sun.   

A straight line PQ touches the ellipse  and the circle x² + y² = r² (b < r < a). RS is a focal chord of the ellipse. If RSis parallel to PQ and meets the circle at points R and S. Find the length of RS. 

If α and β are the eccentric angles of the extremities of a focal chord of an ellipse, then the eccentricity of the ellipse is 

, then the chord joining two points θ₁ and θ₂ on the ellipse  will subtend a right angle at

The number of values of c such that the straight line y = 4x + ctouches the curve , is