Question

Find all points on x + y = 4 that lie at a unit distance from the line 4x + 3y - 10 = 0

Answer 1

Note that the co-ordinates of an arbitrary point on x + y = 4 can be obtained by putting x = t (or y = t) and then obtaining y (or x) from the equation of the line where t is a parameter
Putting x = t in the equation x + y = 4 of the given line we obtain y = 4 - t So co-ordinates of an arbitrary point on the given line are P(t, 4 - t) Let P(t, 4 - t) be the required point Then distance of P from the line 4x + 3y - 10 = 0 is unity i.e.
$\Rightarrow \left|\frac{4t+3(4-t)-10}{\sqrt{4^2+3^2}}\right|=1\Rightarrow |t+2|=5$
$\Rightarrow t+2=\pm 5$ $\Rightarrow$ $t=-7$ or $t=3$.

Hence the required points are $(-7,11)$ or $(3,1)$