**Part I: Multiple Choice Questions:**

*Example 1:*

Locus of the mid-points of the chords of the circle which subtend a right angle at the centre is (a) (b) (c) (d)

*Answer 1: C.*

Solution 1:

Let O be the centre of the circle , and let AB be any chord of this circle, so that . Let be the mid-point of AB. Then, OM is perpendicular to AB. Hence, . Therefore, the locus of is .

*Example 2:*

If the equation of one tangent to the circle with centre at from the origin is , then the equation of the other tangent through the origin is (a) (b) (c) (d) .

*Answer 2: C.*

Solution 2:

Since touches the given circle, its radius equals the length of the perpendicular from the centre to the line . That is,

.

Let be the equation of the other tangent to the circle from the origin. Then,

, which gives two values of m and hence, the slopes of two tangents from the origin, with the product of the slopes being -1. Since the slope of the given tangent is -3, that of the required tangent is 1/3, and hence, its equation is .

*Example 3.*

A variable chord is drawn through the origin to the circle . The locus of the centre of the circle drawn on this chord as diameter is (a) (b) (c) (d) .

*Answer c.*

Solution 3:

Let be the centre of the required circle. Then, being the mid-point of the chord of the given circle, its equation is .

Since it passes through the origin, we have .

Hence, locus of is .

**Quiz problem:**

A line meets the coordinate axes in A and B. A circle is circumscribed about the triangle OAB. If m and n are the distances of the tangent to the circle at the origin from the points A and B respectively, the diameter of the circle is (a) (b) (c) (d) .

*To be continued,*

*Nalin Pithwa.*