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.
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.