Shortest Distance Question

Question

1. Find the shortest distance from point (2, 3) to the line through (–3, 2) and (1, –4). (8 marks)

Please answer in detail

Answer 1

Slope of line: m = (-4 - 2)/(1 + 3) = - 6/4 = - 3/2; ; equation of line: y = 2 - 3/2(x + 3) or y = - 3/2x - 5/2

Now distance function d² = f(x) = (x - 2)² + (- 3/2x - 5/2 - 3)² = x² - 4x + 4 + 9/4x² + 66/4x + 121/4 =

13/4x² + 50/4x + 137/4;

At x = (- 50/4)/(13/2) = - 25/13 the distance will be shortest

min d = √f(-25/13) = 4.715

Answer 2

distance (2, 3) to the line through (–3, 2) and (1, –4).

find line equation and put into y=mx+b form

label (2,3) as (x₀,y₀)

can prove distance d=|mx₀+b-y₀|/√(m²+1)

by using similar right triangles

m=(-4-2)/(1-(-3))=-3/2

(-3/2)(x-1)=y-(-4)

y=-(3/2)x-5/2,,,,,,,check x=-3, y=2

d=|(-3/2)*2-5/2-3|/√((-3/2)²+1)

d=(17/2)/√(13/4)=17/√13