, Then The Chord Joining Two Points θ1 and θ2 on The Ellipse  will Subtend A Right Angle At

Determine the equation of major and minor axes of the ellipse  

Also, find its centre, length of the latusrectum and eccentricity.

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 

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