Find-the-shortest-distance-between-the-curves-9x-2-9y-2-30y-16-0-and-y-2-x-3-

Question Number 37777 by rahul 19 last updated on 17/Jun/18

Find the shortest distance between

the curves 9 x^{2} + 9 y^{2} - 30 y + 16 = 0 a n d

y^{2} = x^{3} .

Answered by MJS last updated on 17/Jun/18

like the similar one you posted earlier

circle : x^{2} + {(y - \frac{5}{3})}^{2} = 1; center C = (\begin{matrix} 0 \\ \frac{5}{3} \end{matrix})

{we}^{'} re looking for the point P = (\begin{matrix} x \\ \pm \sqrt{x^{3}} \end{matrix}) with

∣ C P ∣ minimal

{we}^{'} re looking for the minimum of

f (x) = x^{2} + {(\frac{5}{3} \pm \sqrt{x^{3}})}^{2} = x^{3} + x^{2} \pm \frac{10}{3} \sqrt{x^{3}} + \frac{25}{9}

f^{'} (x) = 3 x^{2} + 2 x \pm 5 \sqrt{x}

3 x^{2} + 2 x - 5 \sqrt{x} = 0 \Rightarrow x_{1} = 0; x_{2} = 1

3 x^{2} + 2 x + 5 \sqrt{x} = 0 \Rightarrow x = 0

f (0) = \frac{25}{9}

f (1) = \frac{73}{9} or \frac{13}{9}

\Rightarrow minimal distance ∣ C P ∣= \frac{\sqrt{13}}{3}

\Rightarrow minimal distance between curves = \frac{\sqrt{13}}{3} - 1

Commented by tanmay.chaudhury50@gmail.com last updated on 17/Jun/18

e x c e l l e n t s i r

Commented by MJS last updated on 17/Jun/18

\frac{\sqrt{13}}{3} - 1 \approx . 20185

. 2 \sqrt{2} \approx . 28284 so this is a good estimate

Answered by tanmay.chaudhury50@gmail.com last updated on 17/Jun/18

Commented by tanmay.chaudhury50@gmail.com last updated on 17/Jun/18

s h o r t e s t d i s t a n c e \dots s e e (0.8, 1) l i e s o n c i r c l e \dots

d i s t a n c e f r o m (0.8, 1) \approx h a l f d i a g o n a l o f t h e

s m a l l e s t s q u r e \frac{1}{2} \times \sqrt{{(0.2)}^{2} + {(0.2)}^{2}} \dots . p l s c o m m e n t