write the equation of the hyperbola with vertices (+-3,0) and asymptotes y= +-4x
write the equation of the sphere with center (1,4,-3) and tangent to the xz plane.
help me out
The equations of the asymptotes are $\displaystyle y+4x=0$ and $\displaystyle y-4x=0$. Multiply these together to get $\displaystyle y^2-16x^2 = 0$. The equation of the hyperbola will then be $\displaystyle y^2-16x^2 = k$, where k is chosen so that the points (±3,0) lie on the curve (namely k=–144). You can then rearrange the equation in the form $\displaystyle \frac{x^2}{3^2} - \frac{y^2}{12^2} = 1$ if you want it to look like a standard hyperbola equation.
(That method will always work. Start by multiplying the equations of the asymptotes together, then change the constant term so that the curve goes through the specified point.)