---
id: 5900f5131000cf542c510025
title: 'Problem 422: Sequence of points on a hyperbola'
challengeType: 5
forumTopicId: 302092
dashedName: problem-422-sequence-of-points-on-a-hyperbola
---

# --description--

Let H be the hyperbola defined by the equation 12x2 + 7xy - 12y2 = 625.

Next, define X as the point (7, 1). It can be seen that X is in H.

Now we define a sequence of points in H, {Pi : i ≥ 1}, as: P1 = (13, 61/4). P2 = (-43/6, -4). For i > 2, Pi is the unique point in H that is different from Pi-1 and such that line PiPi-1 is parallel to line Pi-2X. It can be shown that Pi is well-defined, and that its coordinates are always rational. You are given that P3 = (-19/2, -229/24), P4 = (1267/144, -37/12) and P7 = (17194218091/143327232, 274748766781/1719926784).

Find Pn for n = 1114 in the following format:If Pn = (a/b, c/d) where the fractions are in lowest terms and the denominators are positive, then the answer is (a + b + c + d) mod 1 000 000 007.

For n = 7, the answer would have been: 806236837.

# --hints--

`euler422()` should return 92060460.

```js
assert.strictEqual(euler422(), 92060460);
```

# --seed--

## --seed-contents--

```js
function euler422() {

  return true;
}

euler422();
```

# --solutions--

```js
// solution required
```