Final answer:
The locus of points from which the tangents to an ellipse subserve a right angle at the center is determined by finding the condition that the chord of contact subtends a right angle at the center, which results in a circle as the locus. This circle's radius is related to the semi-major and semi-minor axes of the ellipse.
Step-by-step explanation:
The student is asking about the locus of points from which the tangents to an ellipse subserve a right angle at the centre. This is a typical problem in coordinate geometry related to conic sections, specifically ellipses. The condition for the chord of contact of tangents from point P(x1, y1) to subserve a right angle at the center of the ellipse is given by the equation T = 0, where T is the equation of the chord of contact.
In the case of an ellipse represented by the equation x²/a² + y²/b² = 1, the equation of the chord of contact T for a point P(x1, y1) that lies outside the ellipse is xx1/a² + yy1/b² = 1. For a right angle at the center, we must satisfy the condition of perpendicularity which relies on the idea that the product of slopes of two perpendicular lines is -1.
The slope of the chord connecting two points A and B (where tangents meet the ellipse) will be equal to the slope of the line joining the center to the midpoint of chord AB, which is the average of y-coordinates of A and B divided by the average of x-coordinates of A and B.
Therefore, the condition that this chord subtends a right angle at the center can be found by setting this average slope equal to the negative reciprocal of the slope of the line joining the center to point P. Simplifying this condition, we obtain the locus of point P, which is a circle. This circle has a radius that is related to the semi-major axis and semi-minor axis of the ellipse.