**Solution:**

Let's assume that one of the zeros of the polynomial is "m" and the reciprocal of this zero is "n".

So, we have two zeros of the polynomial: "m" and "n".

**Zero of the Polynomial**
The zero of a polynomial is a value of x for which the polynomial becomes zero. In other words, if we substitute the zero value in the polynomial, the equation will be satisfied.

**Reciprocal of a Number**
The reciprocal of a number is a value that, when multiplied by the original number, gives a product of 1. In other words, if we multiply a number by its reciprocal, we get 1.

**Given Polynomial**
The given polynomial is (a² - 9)x² + 13x + 6a.

**Sum and Product of Zeros**
For a quadratic polynomial in the form of ax² + bx + c, the sum of the zeros is -b/a and the product of the zeros is c/a.

**Sum of Zeros**
The sum of the zeros m and n is given by:
m + n = -b/a

In our case, the sum of the zeros is m + n = -13/(a² - 9).

**Product of Zeros**
The product of the zeros m and n is given by:
mn = c/a

In our case, the product of the zeros is mn = 6a/(a² - 9).

**Reciprocal Relationship**
Since the zero m is the reciprocal of n, we have the relationship:
m = 1/n

**Substituting Values**
Substituting the value of m from the reciprocal relationship into the sum of zeros equation, we have:
1/n + n = -13/(a² - 9)

**Simplifying Equation**
Multiplying both sides of the equation by n(a² - 9), we get:
(a² - 9) + n²(a² - 9) = -13n

Expanding the equation, we have:
a² - 9 + a²n² - 9n² = -13n

**Rearranging Terms**
Rearranging the equation, we have:
a²(1 + n²) - 9(1 + n²) = -13n

**Factoring Out Common Terms**
Factoring out the common term (1 + n²), we have:
(1 + n²)(a² - 9) = -13n

**Dividing Both Sides by (a² - 9)**
Dividing both sides of the equation by (a² - 9), we get:
1 + n² = -13n/(a² - 9)

**Simplifying the Equation**
Multiplying both sides of the equation by (a² - 9), we have:
(a² - 9) + n²(a² - 9) = -13n

**Expanding the Equation**
Expanding the equation, we have:
a² - 9 + a²n² - 9n² = -13n

**Rearranging the Terms**
Rearranging the equation, we have:
a²(1 + n²) - 9(1 + n²) = -13n

**Factoring Out Common Terms**
Factoring out the common term