Question

Find the number of points on the straight line which joins $\left(-4,11\right)$ to $\left(16,-1\right)$ whose co-ordinates are positive integer.

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is C $3$

$\text{formula}\text{used}$
$\text{equation of line passing through}\left({\text{x}}_{\text{1}}{\text{,y}}_{\text{1}}\right)\text{,}\left({\text{x}}_{\text{2}}{\text{,y}}_{\text{2}}\right)$
${\text{y-y}}_{\text{1}}\text{=}\frac{{\text{y}}_{\text{2}}{\text{-y}}_{\text{1}}}{{\text{x}}_{\text{2}}{\text{-x}}_{\text{1}}}\left({\text{x-x}}_{\text{1}}\right)$
$\text{Given}$
$\left(\text{-4,11}\right)\text{and}\left(\text{16,-1}\right)$
$\text{slope}\text{m=}\frac{\text{-1-11}}{\text{16-}\left(\text{-4}\right)}\text{=}\frac{\text{-12}}{\text{20}}\text{=}\frac{\text{-3}}{\text{5}}$
$\text{equation}\text{of}\text{line}$
$\text{y-11=}\frac{\text{-3}}{\text{5}}\left(\text{x-}\left(\text{-4}\right)\right)$
$\text{y-11=}\frac{\text{-3}}{\text{5}}\left(\text{x+4}\right)$
$\text{5y-55=-3x-12}$
$\text{3x+5y=43}$
$\text{at}\text{y=2}$
$\text{3x+5}\left(\text{2}\right)\text{=43}$
$\text{3x=33}$
$\text{x=11}$
$\left(\text{11,2}\right)$
$\text{at}\text{y=5}$
$\text{3x+5}\left(\text{5}\right)\text{=43}$
$\text{3x=18}$
$\text{x=6}$
$\left(\text{6,5}\right)$
$\text{at}\text{y=8}$
$\text{3x+5}\left(\text{8}\right)\text{=43}$
$\text{3x=3}$
$\text{x=1}$
$\left(\text{1,8}\right)$
$\text{Hence}\text{only}\text{3}\text{points}\text{are}\text{possible}$