TIFR CSE 2018 | Part A | Question: 1

Question

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 $A?$

• A. $2$
• B. $6$
• C. $7$
• D. $12$
• E. $14$

Comments

https://math.stackexchange.com/questions/1624364/number-of-chords-having-integral-length

Answer 1

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 : $1 + 1 + 2\times (30 - 24 - 1) = 12.$

Correct Option: D.