How to find the equation of a hyperbola given the centre, focus and vertex?

Hyperbola is a type of curve that has an equation of the form (x - h) / a² - (y - k) / b² = 1, where a, b, h and k are constants, and a,b are not equal to zero. They have many applications in various fields.

Answer: We can find the equation of hyperbola by finding the values of a and b from the given vertex, focus and centre and substitute in the general equation of the hyperbola.

Let's understand the solution in detail.

Explanation:

Let's understand with the help of an example.

Let's say that the center of a given hyperbola is at (-1, -1), focus at (-3, -1), and vertex at (-2, -1).

Now since, the y-coordinates of centre, focus, and vertex are -1, they all lie on the horizontal line y = -1.

General equation of hyperbola is (x - h) / a² - (y - k) / b² = 1, where (h, k) is the centre. Here, h = -1, k = -1.

Further, a is the distance of the vertex from the centre, and c is the distance of focus from the centre.

Hence, from given data, a = 1 and c = 2.

Also, we know that value of b is given by the equation a² + b² = c².

Hence, from the given values of a and c, we have b = √3 from the above equation.

Hence, substituting the values of h, k, a and b, the equation of the hyperbola is: (x + 1)² / 1 - (y + 1)² / 3 = 1.

Hence, we can find the equation of hyperbola by finding the values of a and b from the given vertex, focus and center, and substitute in the general equation of a hyperbola.