Equations of hyperbolas Find an equation of the following hyperbolas, assuming the center is at the origin. Sketch a graph labeling the vertices, foci, and asymptotes. Use a graphing utility to check your work.A hyperbola with vertices (±2, 0) and asymptotes y = ±3x/2

Solution 47EStep 1:In this problem we need to find an equation of the hyperbolas, assuming the center is at the origin.Given A hyperbola with vertices (±2, 0) and asymptotes y = ±3x/2The standard form of the equation of a hyperbola with center (0,0) and transverse axis on the x-axis iswhere* the length of the transverse axis is 2a* the coordinates of the vertices are (±a,0) * the length of the conjugate axis is 2b * the coordinates of the co-vertices are (0,±b)* the distance between the foci is 2c, where * the coordinates of the foci are (±c,0)* the equations of the asymptotes are Step 2:We have vertices: Compare with standard form (±a,0) we get, Given: y = ±3x/2Compare with standard form we get, Thus the equation of the hyperbola is Now let us find fociThe foci of this hyperbola is given by where We have Therefore the foci of the hyperbola is