Solution:

Let's consider the quadratic equation ax^2 + bx + c = 0, where a, b, and c are constants.

The sum of the roots of the quadratic equation can be found using Vieta's formulas. According to Vieta's formulas, the sum of the roots is given by:

Sum of roots = -b/a

Now, let's find the sum of the squares of their reciprocals:

Let the roots of the quadratic equation be r and s.

The reciprocal of a number x is 1/x.

So, the reciprocals of the roots r and s are 1/r and 1/s, respectively.

The sum of the squares of their reciprocals is:

(1/r)^2 + (1/s)^2 = 1/r^2 + 1/s^2

Using the formula for the sum of squares of roots, we can rewrite the above expression as:

(1/r)^2 + (1/s)^2 = (r^2 + s^2)/(r^2s^2)

Now, we have the sum of the roots of the quadratic equation, which is -b/a, equal to the sum of the squares of their reciprocals, which is (r^2 + s^2)/(r^2s^2).

So, we can write:

-b/a = (r^2 + s^2)/(r^2s^2)

Cross multiplying, we get:

-b*r^2*s^2 = a(r^2 + s^2)

Expanding, we get:

- b*r^2*s^2 = a*r^2 + a*s^2

Rearranging the terms, we get:

a*r^2 + b*r^2*s^2 + a*s^2 = 0

Factoring out the common term r^2, we get:

r^2(a + b*s^2) + a*s^2 = 0

Since this equation holds for all values of r and s, the coefficients of each term must be zero. Therefore, we have:

a + b*s^2 = 0
a*s^2 = 0

From the second equation, we can conclude that a = 0 or s = 0.

If a = 0, then the equation becomes 0 + b*r^2*s^2 + 0 = 0, which implies b = 0.

If s = 0, then the equation becomes a*r^2 + 0 + 0 = 0, which implies a = 0.

Therefore, we can conclude that a must be non-zero.

Hence, the correct answer is c. 1.