Question

Let $\alpha$ be the distance between the lines -x + y =2 and x - y = 2, and $\beta$ be the distance between the lines 4x - 3y = 5 and 6y - 8x = 1, then find $\alpha$ and $\beta$

Answer 1

Distance formula is given by,

$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$

Distance between lines $-x+y=2$and $x-y=2$ is,

$d=\sqrt{(\frac{1}{2}-(-\frac{1}{2}){)}^2+((-\frac{1}{2})-\frac{1}{2}{)}^2}$

$d=\sqrt{1+1}$

$\alpha =d=\sqrt{2}$

Distance between lines $4x-3y=5$and $-8x+6y=1$ is,

$d=\sqrt{(-8-\frac{4}{5}{)}^2+(6+\frac{3}{5}{)}^2}$

$d=\sqrt{121}$

$\beta =d=11$