Question

Find the all possible values of $\alpha$ such that point $P(\alpha ,a)$ is outside the parabola $y=x^2+x+1$ and the circle $x^2+y^2=50$

• A. $(-5,\infty )\cup (5,\infty )$
• B. $(-\infty ,\infty )$
• C. $(1,5)$
• D. $(-5,5)$

Answer 1

The correct option is C $(1,5)$

For a Point $P(\alpha ,\alpha )$ to lie outside circle

${\alpha }^2+{\alpha }^2>50\Rightarrow |\alpha |>5$

Parabola $\Rightarrow {(x+\frac{1}{2})}^2=4\left(\frac{1}{4}\right)(y-\frac{3}{4})$

For a Point $P(\alpha ,\alpha )$ to lie outside parabola

${(\alpha +\frac{1}{2})}^2>\alpha -\frac{3}{4}$

$\Rightarrow {\alpha }^2+\alpha +\frac{1}{4}>\alpha -\frac{3}{4}$

$\Rightarrow {\alpha }^2>1$

$\Rightarrow \left|\alpha \right|>1$

So, $\alpha \in (1,5)$