Question

If the point $p\left(a^2,a\right)$ lies in the region. Corresponding in the acute angle between the lines $2y=xand4y=x,$ then find the value of $a$ or the range in which $a$ lies.

Answer 1

$x=2y,4y=x$

$4=x-2y,v=x-4y$

Joint eq’s

$u.v=0$

$\left(x-2y\right)\left(x-4y\right)=0$

$x^2-6yx+8y^2=0$

$s\left(x,y\right)=x^2-6xy+8y^2=0$

If point $p\left(a^2,a\right)$, lies in interior of awte angle formed by these lines,

Then $s\left(x,y\right)$ at $p\left(a^2,a\right)<0$

$\Rightarrow {\left(a^2\right)}^2-+a^2-a+8a^2<0$

$\Rightarrow a^2\left(a^2-6a+8\right)<0$

$\Rightarrow a^2-6a+8<0$

$\Rightarrow \left(a-2\right)\left(a-4\right)<0$

$a\in \left(2,4\right)$