Question

If the roots of the equation
$3a{x}^{2} + 2bx + c = 0$
are in the ratio 2:3, then which is correct

$1) \: 8ac = 25b \\ 2) \: 8ac = 9 {b}^{2} \\ 3) \: 8 {b}^{2} = 9ac \\ 4) \: 8 {b}^{2} = 25ac$

Answer 1

Given Equation is 3ax^2 + 2bx + c = 0.

Here a = 3a, b = 2b, c = c.

Let 2x,3x be the roots of the equation.

Here,

(i)

Sum of roots = -b/a

= > 2x + 3x = -2b/3a

= > 5x = -2b/3a.

= > x = -2b/15a  

(ii)

Product of roots = c/a

= > (2x)(3x) = c/3a

= > 6x^2 = c/3a

= > 6(-2b/15a)^2 = c/3a

= > 6(-4b^2/225a^2) = c/3a

= > (4b^2/225a^2) = c/18a

= > 18a(4b^2) = 225a^2c

= > 18 * 4b^2 = 225ac

= > 72b^2 = 9 * 25ac

= > 9 * 8b^2 = 9 * 25ac

= > 8b^2 = 25ac.

The answer is Option (4). - 8b^2 = 25ac

Hope this helps!