How to calculate total number of chords, if radius of circle given and length of chord is given?

Question

Problem is : There is circle of radius 17 cm and there is a point inside the circle such that its distance is 12 cm from center, How many chords can drawn to this point whose length is an integer? What's the approach? I think infinite, is it right?

Answer 1

HINT: the maximal length is that of a diameter passing through the point and the center of the circle, that is to say $34$ cm. The minimal length is that of the chord perpendicular to this diameter. To calculate the length of this minimal chord, connect its edges with the center and trace the perpendicular from the center to the chord. You get two equal right triangles whose hypothenuse is the radius ($17$ cm) and whose one leg is the distance between the point and the center ($12$ cm). So you get that the length of the minimal chord is $2 \sqrt {17^2-12^2} \approx 24.1 \, \,\,\,$. Now simply count how many integers there are between these two values, reminding that you can start from the maximal chord (i.e. the diameter) and progressively rotate to get decreasing chords in two opposite directions.

Answer 2

Hint

If call that point $P$ then its power is $|12^2-17^2|= 5 \cdot 29$.

If we draw a chord (through $P$) that meet the circle at the points $A$ and $B$ the power of $P$ is also given by $PA\cdot PB$, so

$$PA\cdot PB= 5\cdot 29$$

You want $PA+PB = k \in \Bbb N$ so

$$PA^2-k\cdot PA+5\cdot 29=0$$

Once $PA \in \Bbb R$ then

$$\Delta= k^2-4.5.29 \ge 0 → k \ge \sqrt{4.5.29} \quad (1)$$

But we also know that the chord must be less then diameter and then $$k=PA+PB\le 2.17=34 \quad (2)$$.

So you just have to find the natural values of $k$ respecting the interval given by $(1)$ and $(2)$.

Can you finish?