Find-the-point-s-on-the-graph-of-3x-2-10xy-3y-2-9-closest-to-the-origin-

Question Number 140288 by liberty last updated on 06/May/21

Find the point (s) on the graph

of {3 x}^{2} + 10 xy + {3 y}^{2} = 9 closest to

the origin

Answered by mr W last updated on 06/May/21

w i t h y = x :

3 x^{2} + 10 x^{2} + 3 x^{2} = 9

16 x^{2} = 9

x = \pm \frac{3}{4} \Rightarrow y = \pm \frac{3}{4}

w i t h y = - x :

3 x^{2} - 10 x^{2} + 3 x^{2} = 9

- x^{2} = 9

\Rightarrow n o s o l u t i o n

p o i n t (\frac{3}{4}, \frac{3}{4}) a n d (- \frac{3}{4}, - \frac{3}{4}) a r e

c l o s e s t t o o r i g i n .

Commented by liberty last updated on 06/May/21

why y = x ?

Commented by mr W last updated on 06/May/21

s y m m e t r y o f h y p e r b o l a

Commented by liberty last updated on 06/May/21

Commented by liberty last updated on 06/May/21

oo do you meant like this

Commented by mr W last updated on 06/May/21

y e s

Answered by liberty last updated on 06/May/21

let (x, y) is a point on curve {3 x}^{2} + 10 xy + {3 y}^{2} = 9

closest to the origin . square the

distance from the origin

u = x^{2} + y^{2} . by implicit diff

(1) \frac{du}{dx} = 2 x + 2 y \frac{dy}{dx}

(2) from the second eq

6 x + 10 y + 10 x \frac{dy}{dx} + 6 y \frac{dy}{dx} = 0

we get \frac{dy}{dx} = - \frac{3 x + 5 y}{5 x + 3 y}

substitute to eq (1)

\Rightarrow \frac{du}{dx} = 2 x - 2 y (\frac{3 x + 5 y}{5 x + 3 y}) = 0

\Rightarrow x^{2} = y^{2}; x = \pm y

substituting in the eq of the

graph {16 x}^{2} = 9 \to {\begin{cases} x = \pm \frac{3}{4} \\ y = \pm \frac{3}{4} \end{cases}

Commented by lyubita last updated on 06/May/21

t e p e b e a

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