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TIFR CSE 2018 | Part A | Question: 1

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Consider a point $A$ inside a circle $C$ that is at distance $9$ from the centre of a circle. Suppose you told that there is a chord of length $24$ passing through $A$ with $A$ as its midpoint. How many distinct chords of $C$ have integer length and pass through $A?$

  1. $2$
  2. $6$
  3. $7$
  4. $12$
  5. $14$
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Since $A$ is the midpoint of the chord, the diameter bisects it. The diameter of the circle is $30$ (Using Pythagoras Theorem). It is the shortest chord that passes through $A$ and the longest chord is the diameter. All the integers between $24$ and the diameter i.e. $30$ account for $2$ distinct chords. This is a consequence of Intermediate Value Theorem i.e., the length of the chord is a decreasing function of the smaller of the angles it makes with the diameter. Therefore we have the number of distinct chords as : $1 + 1 + 2\times (30 - 24 - 1) = 12.$

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