Question

There is a point (p,q) on the graph of $f\left(x\right)=x^2$ and a point (r, s) on the graph of $g\left(x\right)=\frac{-8}{x}$ where p > 0, r > 0. If line through (p, q) and (r, s) is also tangent to both the curves at these points respectively, then find the value of $(p+r)$.

Answer 1

$f\left(p\right)=q=p^{2}1>0,r<0$

$g\left(r\right)=s=\frac{-8}{r}\Rightarrow a<0,s<0$

slope of tangent $=f^{\prime }\left(p\right)=g^{\prime }\left(r\right)=$ slope of $(p,q)$ to $(r,s)$

$\Rightarrow 2p=\frac{+8}{r^2}=\frac{s-q}{r-p}$

$p=\frac{4}{r^2}\frac{s-q}{r-p}=2p$

$\Rightarrow \frac{\frac{-8}{r}-p^2}{r-p}=2p$

$\Rightarrow \frac{-8}{r}-p^2=2pr-2p^2$

$\Rightarrow \frac{16}{r}=p^2=\frac{16}{r^4}\Rightarrow =1p=4$

$\therefore (p+r)=5$