---
id: 5900f5001000cf542c510012
title: 'Problem 404: Crisscross Ellipses'
challengeType: 5
forumTopicId: 302072
dashedName: problem-404-crisscross-ellipses
---

# --description--

$E_a$ is an ellipse with an equation of the form $x^2 + 4y^2 = 4a^2$.

$E_a'$ is the rotated image of $E_a$ by $θ$ degrees counterclockwise around the origin $O(0, 0)$ for $0° < θ < 90°$.

<img class="img-responsive center-block" alt="ellipse E_a and ellipse rotated by θ degrees E_a'" src="https://cdn.freecodecamp.org/curriculum/project-euler/crisscross-ellipses.gif" style="background-color: white; padding: 10px;">

$b$ is the distance to the origin of the two intersection points closest to the origin and $c$ is the distance of the two other intersection points.

We call an ordered triplet ($a$, $b$, $c$) a canonical ellipsoidal triplet if $a$, $b$ and $c$ are positive integers.

For example, (209, 247, 286) is a canonical ellipsoidal triplet.

Let $C(N)$ be the number of distinct canonical ellipsoidal triplets ($a$, $b$, $c$) for $a ≤ N$.

It can be verified that $C({10}^3) = 7$, $C({10}^4) = 106$ and $C({10}^6) = 11\\,845$.

Find $C({10}^{17})$.

# --hints--

`crisscrossEllipses()` should return `1199215615081353`.

```js
assert.strictEqual(crisscrossEllipses(), 1199215615081353);
```

# --seed--

## --seed-contents--

```js
function crisscrossEllipses() {

  return true;
}

crisscrossEllipses();
```

# --solutions--

```js
// solution required
```