If the roots of equation of 3ax2 + 2bx + c = 0 are in the ratio 2 : 3 thena)25c = 8b2b)8b2= 25acc)8a2 = 16bcd)6a2 = 13bcCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

Explanation:
The roots of the quadratic equation $ax^2 + bx + c = 0$ are given in the ratio 2:3. Let the roots be 2k and 3k, where k is a constant.

Sum of roots:
The sum of the roots of a quadratic equation is given by -b/a. Therefore,
$$2k + 3k = -\frac{b}{a}$$
$$5k = -\frac{b}{a}$$

Product of roots:
The product of the roots of a quadratic equation is given by c/a. Therefore,
$$2k \times 3k = \frac{c}{a}$$
$$6k^2 = \frac{c}{a}$$

Ratio of the roots:
Given that the roots are in the ratio 2:3, we have:
$$\frac{2k}{3k} = \frac{2}{3}$$
$$2k = \frac{2}{3} \times 3k$$
$$2k = 2k$$

Equating the values of b and c:
From the above equations, we can write:
$$5k = -\frac{b}{a}$$
$$6k^2 = \frac{c}{a}$$
Solving for b and c, we get:
$$b = -5ak$$
$$c = 6ak^2$$

Substitute b and c:
Substitute the values of b and c in the equation $25c = 8b^2$, we have:
$$25(6ak^2) = 8(-5ak)^2$$
$$150ak^2 = 200a^2k^2$$
$$150 = 200a$$
$$3 = 4a$$
$$a = \frac{3}{4}$$
Therefore, the correct answer is option 'B': $8b^2 = 25ac$.

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