Question

If the slope of $OA$ is 1, slope of $OB$ is 7 and $OA=OB$, where $A$ and $B$ are points in the first quadrant and ' $O$ ' is the origin then the absolute value of $10$ times the slope of $AB$ is

Answer 1

Let $OA=OB=r$
Using parametric coordinates,
$A=(0+r\cos \frac{\pi }{4},0+r\sin \frac{\pi }{4})=(\frac{r}{\sqrt{2}},\frac{r}{\sqrt{2}})$
$B=(0+r\cos \alpha ,0+r\sin \alpha )=(r\cos \alpha ,r\sin \alpha )$ where $\tan \alpha =7.$
Now Slope of $AB=\frac{r\sin \alpha -r/\sqrt{2}}{r\cos \alpha -r/\sqrt{2}}$
$=\frac{7/\sqrt{5}0-1/\sqrt{2}}{1/\sqrt{5}0-1/\sqrt{2}}$
$=\frac{-1}{2}$

Now absolute value of $10$ times the slope $=|10\times \frac{-1}{2}|=5$