**Solution:**

Let the first number be x

=> second number = 27 – x.

Reciprocal of first number = 1/x

Reciprocal of second number = 1/(27-x)

According to the given condition,

=> 36 = x(27 – x)

=> 36 = 27x – x^{2}

=> x^{2} – 27x + 36 = 0

=>the required quadratic equation is x^{2} – 27x + 36 = 0.