Question

If points $(-7,5)\text{and}(\alpha ,{\alpha }^2)$ lie on the opposite sides of the line $5x-6y-1=0$ then

• A. $\alpha \in [0,1]$
• B. $a\in [-1,0]$
• C. $\alpha \in (\frac{1}{3},\frac{1}{2})$
• D. $\alpha \in [2,4]$

Answer 1

The correct option is C $\alpha \in (\frac{1}{3},\frac{1}{2})$

$(-7,5)$ & $(\alpha ,{\alpha }^2)$ lie on opposite side of $5x-6y-1=0$

$(-7,5)\to 5x-6y-1=5(-7)-6\left(5\right)-1$

$=-35-30-1$

$=-66<0$

as $(\alpha ,{\alpha }^2)$ should be on the opposite of $5x-6y-1=0$

$\Rightarrow (\alpha ,{\alpha }^2)\to 5x-6y-1=5\alpha -6{\alpha }^2-1>0$

$\Rightarrow 6{\alpha }^2-5\alpha +1<0$

$\Rightarrow 6{\alpha }^2-2\alpha -3\alpha +1<0$

$\Rightarrow 2\alpha (3\alpha -1)-1(3\alpha -1)<0$

$\Rightarrow (2\alpha -1)(3\alpha -1)<0$

$\Rightarrow \frac{1}{3}<\alpha <\frac{1}{2}$.