Find the condition to be satisfied by the coefficients of the equation p x2+q x+r=0, so that the roots are in the ratio 3: 4.

Question

Find the condition to be satisfied by the coefficients of the equation p x2+q x+r=0, so that the roots are in the ratio 3: 4. (A) 12 q2=49 p r (B) 12 q2=-49 p r (C) 49 q2=12 pr (D) 49 q2=-12 p r

Tardigrade Admin · Accepted Answer

Let the common factor for the roots be &gamma;. The roots are 3 y and 4 y, 3 y+4 y=(-q/p) ⇒ y=(-q/7 p) (3 y)(4 y)=(r/p) ⇒ y2=(r/12 p) &gamma;2=((-q/7 p))2=(r/12 p) 12 q2=49 p r

Q. Find the condition to be satisfied by the coefficients of the equation $p x^{2} + q x + r = 0$ , so that the roots are in the ratio $3 : 4$ .

$12 q^{2} = 49 p r$

$12 q^{2} = - 49 p r$

$49 q^{2} = 12 p r$

$49 q^{2} = - 12 p r$

Let the common factor for the roots be $γ$ .
The roots are $3 y$ and $4 y, 3 y + 4 y = \frac{- q}{p}$ $\Rightarrow y = \frac{- q}{7 p}$
$(3 y) (4 y) = \frac{r}{p} \Rightarrow y^{2} = \frac{r}{12 p}$
$γ^{2} = (\frac{- q}{7 p})^{2} = \frac{r}{12 p} 12 q^{2} = 49 p r$

Q. Find the condition to be satisfied by the coefficients of the equation px2+qx+r=0, so that the roots are in the ratio 3:4.

12q2=49pr

12q2=−49pr

49q2=12pr

49q2=−12pr