Question

Prove that the curves $x=y^2$ and $xy=k$ cut at right angles, if $8k^2=1$

Answer 1

$f\left(x\right)=\sqrt{x}$

$g\left(x\right)=\frac{k}{x}$

$\frac{k}{x}=\sqrt{x}$

$x\sqrt{x}=k$

$x^3=k^2$

$x=k^{\frac{2}{3}}$

$f\left(x_0\right)g^1\left(x_0\right)=-1$

$\Rightarrow \frac{1}{2\sqrt{x_0{x}^2}}=1\Rightarrow \frac{k}{{\sqrt{k}}^{\frac{2}{3}}x{\left(k^{\frac{2}{3}}\right)}^2}=2$

$\frac{k}{k^{\frac{1}{3}}{k}^{\frac{4}{3}}}=2$

$k^{\frac{2}{3}}=2$

$2k^{\frac{2}{3}}=1$

$8k^2=1$