(a) Let $(x_0, y_0)$, where $y \neq 0$ be a point on the hyperbola $\frac{x^2} {a^2} - \frac{y^2} {b^2} = 1$. Find an equation of the tangent line to the hyperbola passing through $(x_0, y_0)$.
I thought this was a typical implicit differentiation problem, but I could not get the suggested answer.
My working
$$\frac {2x} {a^2} - \frac {dy} {dx} \frac {2y} {b^2} = 0$$
$$\implies \frac {dy} {dx} = \frac {b^2x} {a^2y}$$
Equation of tangent is $y = \frac {b^2x_0} {a^2y_0} (x - x_0) + y_0$.
Suggested answer
Equation of tangent is $\frac{x_0x} {a^2} - \frac {y_0y} {b^2} = 1$.
(b) Let $(x_0, y_0)$, where $y \neq 0$ be a point on the parabola $y^2 = 2px$. Find an equation of the tangent line to the parabola passing through $(x_0, y_0)$.
Similarly, I thought this was a typical implicit differentiation problem, but I also could not get the suggested answer.
My working
$$2y\frac {dy} {dx} = 2p$$
$$\implies \frac {dy} {dx} = \frac p y$$
Equation of tangent is $y = \frac {p} {y_0} (x - x_0) + y_0$.
Suggested answer
Equation of tangent is $y_0y = p(x + x_0)$.
The suggested answers are correct, so I must have done something wrong for both parts, though I know not what. Moreover, I have a hunch that it is the same mistake that I have repeated. Anyone kind enough to point out where I have gone wrong will be greatly appreciated :)