Using linear algebra, it is possible to prove that the shortest distance between a point (ro,...

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Using linear algebra, it is possible to prove that the shortest distance between a point (ro, yo) and a line with equation ax +by+c=0 is ✓d= However, a calculus proof of this result is also possible using param- etrized curves. laxo+byo + cl √a² +6² (a) Show that X(t) = (-bt - a/c, at) with tER is a parametriza- tion of the line with equation ax +by+ c = 0. (b) Find the value to of t that minimizes the function f(t) = ||X(t) - (xo, yo) || = √√(-bt-a/c-xo)² + (at-yo)², which gives the distance between a point X(t) on the line and the point (xo, yo). 107 (c) Show that f(to) simplifies to the distance formula given above. 1 Using linear algebra, it is possible to prove that the shortest distance between a point (ro, yo) and a line with equation ax +by+c=0 is ✓d= However, a calculus proof of this result is also possible using param- etrized curves. laxo+byo + cl √a² +6² (a) Show that X(t) = (-bt - a/c, at) with tER is a parametriza- tion of the line with equation ax +by+ c = 0. (b) Find the value to of t that minimizes the function f(t) = ||X(t) - (xo, yo) || = √√(-bt-a/c-xo)² + (at-yo)², which gives the distance between a point X(t) on the line and the point (xo, yo). 107 (c) Show that f(to) simplifies to the distance formula given above. 1

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ANSWER As we chose P to be arbitary point it ... View the full answer