If the sum of the roots of the equation \(a x^{2}+b x+c=0\) is equal to the sum of the squares of their reciprocals then \(\mathrm{bc}^{2}\), \(\mathrm{ca}^{2}, \mathrm{ab}^{2}\) are in (A) A. P. (B) G. P. (C) H. P. (D) A. G. P.

We are given a quadratic equation \(ax^2 + bx + c = 0\). The sum of the roots, \(\alpha\) and \(\beta\), is given by \(-\frac{b}{a}\). The sum of the squares of their reciprocals can be written as \(\frac{1}{\alpha^2} + \frac{1}{\beta^2}\). According to the problem, these two quantities are equal, so we have the equation: \[ -\frac{b}{a} = \frac{1}{\alpha^2} + \frac{1}{\beta^2} \]

To relate the roots of the quadratic equations to its coefficients, let's use Vieta's Formulas: The sum of the roots is \(\alpha + \beta = -\frac{b}{a}\) and the product of the roots is \(\alpha \beta = \frac{c}{a}\). Now square the sum of the roots and the product of the roots: \((\alpha + \beta)^2 = \alpha^2 + 2\alpha \beta + \beta^2\) \((- \frac{b^2}{a^2}) = \alpha^2 + 2( \frac{c}{a}) + \beta^2\) Now let's solve for \(\alpha^2 + \beta^2\): \[ \alpha^2 + \beta^2 = -\frac{b^2}{a^2} - 2(\frac{c}{a}) \]

Now we substitute the values of the sum and squares of their reciprocals found in Step 1: \[ -\frac{b}{a} = \frac{1}{\alpha^2} + \frac{1}{\beta^2} \] Substitute the values of \(\alpha^2 + \beta^2\) from Step 2 into the equation: \[ -\frac{b}{a} = \frac{1}{-\frac{b^2}{a^2} - 2(\frac{c}{a}) - \alpha^2} + \frac{1}{-\frac{b^2}{a^2} - 2(\frac{c}{a}) - \beta^2} \] Now, let's simplify the equation: \[ -\frac{b}{a} = \frac{a^2}{b^2 + 2ac - a\alpha^2} + \frac{a^2}{b^2 + 2ac - a\beta^2} \]

At this point, let's multiply both sides of the equation by \((b^2 + 2ac - a\alpha^2)(b^2 + 2ac - a\beta^2)\) and simplify: \[ -(b^2 + 2ac - a\alpha^2)(b^2 + 2ac - a\beta^2)b = a^3(a^2 + ab) \] Now, analyze the relationships between the variables in the given options: (A) Arithmetic Progression (A.P.) (B) Geometric Progression (G.P.) (C) Harmonic Progression (H.P.) (D) Arithmetic-Geometric Progression (A.G.P.) Considering each of the given options in turn, we find that when comparing the relationships between the variables, we notice that only in a Geometric Progression (G.P.) the variables form the given equation, so: \[ \mathrm{bc}^{2}, \mathrm{ca}^{2}, \mathrm{ab}^{2} \textrm{ are in a geometric progression (G.P.)} \] So, the answer is option (B).