Let-P-be-point-on-the-graph-of-a-straight-line-y-2x-3-and-Q-be-a-point-on-the-graph-of-a-parabola-y-x-2-x-1-Find-the-shortest-distance-between-P-and-Q-

Question Number 126605 by bramlexs22 last updated on 22/Dec/20

L e t P b e p o i n t o n t h e g r a p h

o f a s t r a i g h t l i n e y = 2 x - 3 a n d Q

b e a p o i n t o n t h e g r a p h o f a p a r a b o l a

y = x^{2} + x + 1 . F i n d t h e s h o r t e s t

d i s t a n c e b e t w e e n P a n d Q .

Answered by liberty last updated on 22/Dec/20

l e t y = 2 x + p i s t a n g e n t l i n e a t c u r v e

y = x^{2} + x + 1 \Rightarrow w e g e t 2 x + 1 = 2; x = \frac{1}{2}

t h e p o i n t Q (\frac{1}{2}, \frac{7}{4}) a n d p = \frac{3}{4}

t h u s ∣ P Q ∣_{m i n} = ∣ \frac{\frac{3}{4} - (- 3)}{\sqrt{5}} ∣ = \frac{15}{4 \sqrt{5}} = \frac{3}{4} \sqrt{5}

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