Geometry - Chords of a circle

A circle of certain radius has a chord in it. With the given parameters, we need to find the minimum value of another chord.

1. Geometry Q18

2. Qn : Chords of a circle ( a) 6  3 ( b ) 4 3 (c) 8  3 (d) 2  3 A circle of radius 5 cm has chord RS at a distance of 3 units from it. Chord PQ intersects with chord RS at T such that TS = of RT. Find minimum value of PQ.  

3. Soln : Chords of a circle OM = 3, OS = 5 A circle of radius 5 cm has chord RS at a distance of 3 units from it. Chord PQ intersects with chord RS at T such that TS = of RT. Find minimum value of PQ.   P Q O T S R M

4. Soln : Chords of a circle MS = 4 = RM {Using Pythagoras theorem} RS = 8 cms TS = of RT TS = of RS If RS = 8 cms TS = 2 cms   A circle of radius 5 cm has chord RS at a distance of 3 units from it. Chord PQ intersects with chord RS at T such that TS = of RT. Find minimum value of PQ.  

5. Soln : Chords of a circle RT × TS = PT × TQ { Intersecting Chords theorem : When there are two intersecting chords, the product of the rectangle formed by the segments of one chord is equal to the product of the rectangle formed by the segments of the other.} 6 × 2 = PT × TQ PT × TQ = 12 By AM – GM inequality, ≥   A circle of radius 5 cm has chord RS at a distance of 3 units from it. Chord PQ intersects with chord RS at T such that TS = of RT. Find minimum value of PQ.  

6. Soln : Chords of a circle ≥ PT + TQ ≥ 2 12  PQ ≥ 2 12 Or PQ ≥ 4 3 Minimum PQ = 4  3 Answer choice (b)   A circle of radius 5 cm has chord RS at a distance of 3 units from it. Chord PQ intersects with chord RS at T such that TS = of RT. Find minimum value of PQ.