Question

If point $(k,k^2)$ lies inside the region bounded by parabolas $y^2=64x$ and $-x^2+x-1+y=0$ then $k$ lies in the interval

• A. $k\in (1,4)$
• B. $k\in (0,4)$
• C. $k\in (1,\infty )$
• D. $k\in (-\infty ,4)$

Answer 1

The correct option is A $k\in (1,4)$

Point$(k,k^2)$ should lie inside both the parabolas
for $y^2=64x$
$(k^2{)}^2-64k<0$
$k(k^3-64)<0$
$k(k-4)(k^2+4k+4^2)<0$
$k\in (0,4)\cdots \left(1\right)$

For $-x^2+x-1+y=0$
$\Rightarrow x^2-x+1-y=0$
Now,
$k^2-k+1-k^2<0$
$k>1\cdots \left(2\right)$
From $\left(1\right)$ and $\left(2\right)$, we get
$k\in (1,4)$