Equation of a hyperbola Consider a hyperbola to be the set of points in a plane whose distances from two fixed points have a constant difference of 2a or -2a. Derive the equation of a hyperbola. Assume the two fixed points are on the x-axis equidistant from the origin.
Solution 86AEStep 1:In this problem we have to derive the equation of hyperbola by using the definition that the differences of the distances between two fixed points is .Given : The two fixed points are on the x-axis equidistant from the origin.the major axis is horizontal.We know that, A hyperbola is "the set of all points in a plane such that the difference of the distances from two fixed points (foci) is constant".When the major axis is horizontal, the foci are at (-c,0) and at (0,c).Therefore the sum of the distances to the point on the hyperbola from the two foci (c,0) and (-c,0) is a 2a.If we let d1 and d2 bet the distances from the foci to the point, then |d1 - d2 |= 2a.The absolute value is around the difference so that it is always positive.Let d1 be the distance from the focus at (-c,0) to the point at (x,y). Since this is the distance between two points, we'll need to use the distance formula.Similarly, d2 will involve the distance formula and will be the distance from the focus at the (c,0) to the point at (x,y).