Poitns A and B are in the first quadrant let ‘O’ be the origin. If slope of OA is 1,  slope of OB is 7 and OA=OB, then slope of AB is

Let  $A=\left(a,b\right),B=\left(c,d\right)$

Slope of OA=1

$\frac{b}{a}=1\Rightarrow a=b$

Slope of OB=7

$\frac{d}{c}=7\Rightarrow d=7c$

$OA=OB\Rightarrow {\left(OA\right)}^2={\left(OB\right)}^2\Rightarrow a^2+b^2=c^2+d^2$

$\Rightarrow a^2+a^2=c^2+49c^2$

$\Rightarrow 2a^2=50c^2$

$\Rightarrow a=5c$

Slope of 

$AB=\frac{d-b}{c-a}=\frac{7c-5c}{c-5c}=\frac{2c}{-4c}=\frac{-1}{2}$