If a, b and c \; (a \gt 0, c \gt 0) are in GP, then consider the following in respect of the equation ax^{2}+bx+c=0: 1. The equation has imaginary roots. 2. The ratio of the roots of the equation is 1:\omega where \omega is a cube root of unity. 3. The product of roots of the equation is (\frac{b^{2}}{a^{2}}). Which of the statements given above are correct?

Explanation

Since a, b, c are in GP, b^2 = ac. The discriminant is \Delta = b^2 - 4ac = ac - 4ac = -3ac. Because a \gt 0 and c \gt 0, -3ac \lt 0, so the roots are imaginary (Statement 1 is true). The roots are x = \frac{-b \pm i\sqrt{3ac}}{2a}. Since \sqrt{ac} = b, x = \frac{b}{a}(\frac{-1 \pm i\sqrt{3}}{2}) = \frac{b}{a}\omega and \frac{b}{a}\omega^2. Their ratio is \omega : \omega^2 = 1 : \omega (Statement 2 is true). The product of the roots is \frac{c}{a} = \frac{ac}{a^2} = \frac{b^2}{a^2} (Statement 3 is true).

Related questions on Algebra

• How many four-digit natural numbers are there such that <strong>ALL</strong> of the digits are odd?
• What is \sum_{r=0}^{n}2^{r}C(n,r) equal to ?
• If different permutations of the letters of the word 'MATHEMATICS' are listed as in a dictionary, how many words (with or without meaning) a...
• Consider the following statements : 1. If f is the subset of Z\times Z defined by f=\{(xy,x-y);x,y\in Z\}, then f is a function from...
• For how many quadratic equations, the sum of roots is equal to the product of roots?