Final answer:
The question pertains to the locus of the center of a circle, the diameter of which is the chord of another circle that passes through the origin. The given circle has the equation x² + y² – 2ax = 0. After employing the midpoint formula, we find that the locus of the center of the smaller circle is given by x² + y² – ax = 0.
Step-by-step explanation:
The subject of your query is related to geometry, especially the geometry of circles and ellipses. As a quick refresher, a circle is a special case of an ellipse where the two foci coincide, i.e., congregate at one point which we term as the 'center'. The distance from this central point to any place on the circle remains constant and this invariable length is known as the radius of the circle. The given equation, x² + y² – 2ax = 0, represents a circle with the center shifted from the origin along the x-axis.
A 'variable chord' refers to a line segment whose end points lie on the circle and passes through the origin. If another circle is drawn using this chord as its diameter, the center of this new circle will be the midpoint of the chord. The trajectory traced by this center as the chord varies will produce a certain path known as the 'locus'.
To find the locus of the center of the second (new) circle, we can use the midpoint formula that provides the coordinates of the midpoint (center) based on the end points of the chord (origin and point on the circle). Based on this computation, the correct option is (2) x² + y² – ax = 0. This equation represents the locus of the center of the smaller circle, which is drawn using the chord passing through the origin of the larger circle as its diameter.
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