The-closest-distance-from-the-point-on-the-ellipse-2x-2-y-2-8-to-the-line-y-5x-is-

Question Number 130213 by benjo_mathlover last updated on 23/Jan/21

The closest distance from the point on

the ellipse {2 x}^{2} - y^{2} = 8 to the line y = 5 x

is__

Answered by liberty last updated on 23/Jan/21

gradient tangent line of

ellipse : 4 x - {2 yy}^{'} = 0

\Rightarrow y^{'} = \frac{2 x}{y} equals to 5; we get 5 y = 2 x

substitute into eq of ellipse

{2 x}^{2} - {(\frac{2 x}{5})}^{2} = 8; {46 x}^{2} = 8 \times 25

x = \pm \frac{10}{\sqrt{23}} \Rightarrow y = \pm \frac{4}{\sqrt{23}}

Now equation of tangent line

the ellipse is 5 x - y = \frac{46}{\sqrt{23}} or

5 x - y = 2 \sqrt{23}, so the closest

distance is required equals to

d = \frac{2 \sqrt{23}}{\sqrt{26}}

Commented by liberty last updated on 23/Jan/21

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