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 Incoming links: Distance from a point to a line

The line has equation

2 x - y + 2 = 0

, with normal vector

(2, - 1)

d = \frac{∣2 ( 4 ) - ( - 6 ) + 2∣}{2 ^{2} + ( - 1 ) ^{2}} = \frac{16}{5} .

(4, - 6) - (16/5) (2, - 1) = (- 12/5, - 14/5)

, which lies on the line and realizes this shortest distance.

1

centred at

(2, 1)

. A line through the origin with equation

k x - y = 0

intersects it precisely when the perpendicular distance from that centre is at most the radius. The distance from a point to a line gives

\frac{∣2 k - 1∣}{k ^{2} + 1} \leq 1 ⟺ (2 k - 1)^{2} \leq k^{2} + 1 ⟺ k (3 k - 4) \leq 0.

Hence the full parameter range is

0 \leq k \leq \frac{4}{3} .

The two endpoint lines are tangent, at

(2, 0)

and

(6/5, 8/5)

respectively; interior values give two real intersections. Both endpoints are therefore included.