One of the independent variables in a logistic regression is the ratio of other two inputs. Is this okay? Suppose that I have a logistic regression with continuous independent variables $a$,$b$, and $c$. In my logistic regression, $c = \frac{a}{b}$. 
Is it all right to include variable $c$ in the logistic regression? 
These independent variables are financial ratios. 
Output of logistic regression using SPSS:
          EXP(coef)
    a      4.937568
    b      1223.406
    c      0.30749
 constant  2.45432

I think that this holds: EXP(coef) value of variable c is lower than 1 (negative effect on dependent variable-odd ratio) because EXP(coef) value for b is much higher than a.
(All coefficients are statistically significant at 5%)
Is this reasoning correct?
 A: Using $a, b, c$ to denote predictors is unconventional notation; I will compromise with you to discuss $x_a$, $x_b$, $x_c$. 
In general, this could be problematic as $x_c \equiv x_a / x_b $ will be correlated with both $x_a$ and $x_b$ in principle and in practice. How serious this will be for you will depend on your data but a correlation matrix for those variables will quantify and a scatterplot matrix will clarify. Having correlations between predictors is not fatal: if it were, most applications of regression would fail at the outset. However, including correlated predictors together always requires care. 
Any kind of ratio can be problematic in regression, indeed statistics generally. Guessing first that $x_a, x_b > 0$ the range $x_a > x_b$ becomes the large extent $x_c > 1$ whereas the opposite $x_a < x_b$ becomes $0 < x_c < 1$. If either variable could be zero or negative these comments require modification, but the stretching and the squeezing can both be awkward or worse. 
We need to know more about the scientific justification for your choices and on the distributions and correlations you have in practice. It is often my experience that when researchers start using ratios they would be better off using logarithms any way. Observing that $\log (x_a / x_b) \equiv \log x_a - \log x_b$ allows us to speculate that using a ratio is often related just to using the logged variables as predictors by themselves. Indeed, once we have $\log x_a$ and $\log x_b$ as predictors, then $\log x_c$ would be redundant. 
Sometimes there is just a scientific convention for use of some ratio measures. A common example in macroeconomics is that GDP, population size and GDP per head are all used descriptively, but including all of those in a regression would usually just produce an analytical mess. 
I've not addressed your specific use of logistic regression here; I don't think using logistic makes this question either easier or more difficult; the nub of the matter is the mix that is $X\beta$, i.e. your matrix of predictors $X$ and the parameter vector $\beta$. 
