The problem states that d, e, f are in Geometric Progression (G.P.) and the two quadratic equations ax² + 2bx + c = 0 and dx² + 2ex + f = 0 have a common root.

Step 1: Since d, e, f are in G.P., we can write e² = df.

Step 2: Let's assume that the common root of the two quadratic equations is 'p'.

Step 3: By Vieta's formulas, the sum of the roots of the first equation is -2b/a and for the second equation is -2e/d. Since 'p' is a common root, we can write the sum of the roots of the first equation as p + q = -2b/a and the sum of the roots of the second equation as p + r = -2e/d, where q and r are the other roots of the first and second equation respectively.

Step 4: Subtracting these two equations, we get q - r = -2b/a + 2e/d.

Step 5: Similarly, by Vieta's formulas, the product of the roots of the first equation is c/a and for the second equation is f/d. We can write the product of the roots of the first equation as pq = c/a and the product of the roots of the second equation as pr = f/d.

Step 6: Subtracting these two equations, we get pq - pr = c/a - f/d.

Step 7: From steps 4 and 6, we can see that the difference of the roots and the difference of the products of the roots of the two equations are proportional to each other, i.e., (q - r)/(pq - pr) = -2b/a + 2e/d / c/a - f/d.

Step 8: Simplifying this equation, we get (q - r)/(pq - pr) = 2(ad - bc) / (ac - df).

Step 9: Since the left side of this equation is a constant, the right side must also be a constant. Therefore, ad - bc = k1 and ac - df = k2, where k1 and k2 are constants.

Step 10: From step 1, we know that e² = df. Substituting this into the equation ac - df = k2, we get ac - e² = k2.

Step 11: Therefore, the required condition for the two quadratic equations to have a common root is that the coefficients a, b, c, d, e, f must satisfy the equations ad - bc = k1 and ac - e² = k2, where k1 and k2 are constants.

Question

The problem states that d, e, f are in Geometric Progression (G.P.) and the two quadratic equations ax² + 2bx + c = 0 and dx² + 2ex + f = 0 have a common root.

Step 1: Since d, e, f are in G.P., we can write e² = df.

Step 2: Let's assume that the common root of the two quadratic equations is 'p'.

Step 3: By Vieta's formulas, the sum of the roots of the first equation is -2b/a and for the second equation is -2e/d. Since 'p' is a common root, we can write the sum of the roots of the first equation as p + q = -2b/a and the sum of the roots of the second equation as p + r = -2e/d, where q and r are the other roots of the first and second equation respectively.

Step 4: Subtracting these two equations, we get q - r = -2b/a + 2e/d.

Step 5: Similarly, by Vieta's formulas, the product of the roots of the first equation is c/a and for the second equation is f/d. We can write the product of the roots of the first equation as pq = c/a and the product of the roots of the second equation as pr = f/d.

Step 6: Subtracting these two equations, we get pq - pr = c/a - f/d.

Step 7: From steps 4 and 6, we can see that the difference of the roots and the difference of the products of the roots of the two equations are proportional to each other, i.e., (q - r)/(pq - pr) = -2b/a + 2e/d / c/a - f/d.

Step 8: Simplifying this equation, we get (q - r)/(pq - pr) = 2(ad - bc) / (ac - df).

Step 9: Since the left side of this equation is a constant, the right side must also be a constant. Therefore, ad - bc = k1 and ac - df = k2, where k1 and k2 are constants.

Step 10: From step 1, we know that e² = df. Substituting this into the equation ac - df = k2, we get ac - e² = k2.

Step 11: Therefore, the required condition for the two quadratic equations to have a common root is that the coefficients a, b, c, d, e, f must satisfy the equations ad - bc = k1 and ac - e² = k2, where k1 and k2 are constants.

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