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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.
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