Sol. (i) Since,
i.e. (distance of from the point )

+ (distance of from the point (constant)
i.e. The sum of the distances of from two fixed points and is constant, which is the definition of an ellipse.
Therefore, locus of satisfying the given condition will be an ellipse with foci at and and centre at origin.
(ii) Given that,

or ......(i)

Let the points and have affixes and and the point has affix .

Then, Eq. (i) can be written as

Thus, locus of is such that the angle subtended at by the line joining points and is .
This is the definition of a circle with diameter .

Therefore, locus of point is a circle with diameter and centre at origin with radius 1 .
(iii) We have, represents a circle with centre at ( 2,3 ) and radius .

Since, represents the region in the plane outside the circle.

........(i)

and represents the region inside circle.

........(ii)

Hence, represent the angular space between the two circles (i) and (ii).
(iv) We have,

Let

The given inequality can be written as

This inequality represents the region between the lines

(v) We have,

Hence, the inequality represents exterior of a circle of radius 10 with center at .

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