Equations $ ax^2 + bx + c = 0 $ and $ cx^2 + bx + a = 0 $ have a common root and the dirfference of other roots is 1. Maximum value of $ | \dfrac {a}{c} | $ is :

Let the common root be alpha, and the other roots be beta and gamma. The given condition states that beta - gamma = 1. Multiplying the respective differences of roots for both equations gives (alpha - beta)(alpha - gamma) = (c - a)/c multiplied by (a - c)/a, which simplifies to -(a - c)^2 / ac. Expanding (alpha - beta)(alpha - gamma) results in alpha^2 - alpha(beta + gamma) + beta gamma. Using the sum and product relations from the two equations to substitute this, we get the maximum value of |a/c| as 1. The result is 1.

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