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.