An ellipse slides between two perpendicular straight lines. Then, the locus of its centre is a/an
Clearly, P is the point of intersection of perpendicular tangents. So, P lies on the director circle of the given ellipse . This means that the centre of the ellipse will always remain at a constant from P. Hence, the locus of C is a circle.
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 locus of the point of intersection of tangents to an ellipse at two points, sum of whose eccentric angles is constant is a/an
The number of values of c such that the straight line y = 4x + ctouches the curve , is
, then PF1 + PF2 equals
The sum of the squares of the perpendicular on any tangent to the ellipse from two points on the minor axis, each at a distance from the centre is