数学一道线段难题Findtheshortestdistancefromthepoint(1,3)totheliney=x-4,tothenearesttenth.请详细解答.多谢.

Question

数学 一道线段难题
Find the shortest distance from the point (1,3) to the line y=x-4, to the nearest tenth.
请详细解答.多谢.

Answer 1

Find the shortest distance from the point A(1,3) to the line y=x-4,to the nearest tenth.
Solution:
Assume point P(x,y) on the line y = x - 4
So,the distance between A(1,3) and P(x,y)
L = √[(x-1)²+(y-3)²]
= √[(x-1)²+(x-4-3)²]
= √[(x-1)²+(x-7)²]
If L has a smallest value,L² also has its smallest value.
L² = (x-1)²+(x-7)²
= 2x² - 16x + 50
= 2(x² - 8x + 25)
= 2[(x² - 8x + 16) + 9]
= 2[(x - 4)² + 9]
So,when x = 4.L²(min) = 18,Lmin = 3√2 ≈ 3.2 (unit)