Find real constants $c$ and $k$ such that $y=cx^k$ passes through point $(a, b)$ with slope $m$

Question

In the Cartesian plane, can a power function of the form $y=cx^k$ (where $c>0$ and $k>1$, not necessarily an integer) be found such that its graph passes through any arbitrary point $(a, b)$ with arbitrary slope $m$? [Assume that $a>0$, $b>0$, and $m>(b/a)$.] If so, how can $c$ and $k$ be calculated in terms of $a$, $b$, and $m$?

Note that this technique could be used to generate a type of spline curve that is tangent to each side of an obtuse angle, such as the one formed by the intersection of the lines $y=y_0$ and $y=mx$ (where $y_0>0$ and $m>0$).

Answer 1

The equations you get are

$$b=ca^k$$ $$m=cka^{k^-1}$$ dividing the first equation by the second yields $$\frac{b}{m}=\frac{a}{k}$$ which can be solved for $k$, and you can then solve for c.

Answer 2

Let $f(x)=cx^k$. Then 1) $f(a)=b$ and 2) $f'(a)=m$