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

The closest distance from the point on the ellipse 2x^2 −y^2 =8 to the line y=5x is __

$$\:\mathrm{The}\:\mathrm{closest}\:\mathrm{distance}\:\mathrm{from}\:\mathrm{the}\:\mathrm{point}\:\mathrm{on}\: \\ $$$$\mathrm{the}\:\mathrm{ellipse}\:\mathrm{2x}^{\mathrm{2}} −\mathrm{y}^{\mathrm{2}} =\mathrm{8}\:\mathrm{to}\:\mathrm{the}\:\mathrm{line}\:\mathrm{y}=\mathrm{5x} \\ $$$$\mathrm{is}\:\_\_ \\ $$

Answered by liberty last updated on 23/Jan/21

gradient tangent line of ellipse : 4x−2yy′=0 ⇒y′=((2x)/y) equals to 5 ; we get 5y=2x substitute into eq of ellipse 2x^2 −(((2x)/5))^2 =8 ; 46x^2 = 8×25 x=± ((10)/( (√(23)))) ⇒y=±(4/( (√(23)))) Now equation of tangent line the ellipse is 5x−y=((46)/( (√(23)))) or 5x−y=2(√(23)) , so the closest distance is required equals to d = ((2(√(23)))/( (√(26))))

$$\mathrm{gradient}\:\mathrm{tangent}\:\mathrm{line}\:\mathrm{of}\: \\ $$$$\mathrm{ellipse}\::\:\mathrm{4x}−\mathrm{2yy}'=\mathrm{0}\: \\ $$$$\Rightarrow\mathrm{y}'=\frac{\mathrm{2x}}{\mathrm{y}}\:\mathrm{equals}\:\mathrm{to}\:\mathrm{5}\:;\:\mathrm{we}\:\mathrm{get}\:\mathrm{5y}=\mathrm{2x} \\ $$$$\mathrm{substitute}\:\mathrm{into}\:\mathrm{eq}\:\mathrm{of}\:\mathrm{ellipse}\: \\ $$$$\mathrm{2x}^{\mathrm{2}} −\left(\frac{\mathrm{2x}}{\mathrm{5}}\right)^{\mathrm{2}} =\mathrm{8}\:;\:\mathrm{46x}^{\mathrm{2}} \:=\:\mathrm{8}×\mathrm{25} \\ $$$$\:\mathrm{x}=\pm\:\frac{\mathrm{10}}{\:\sqrt{\mathrm{23}}}\:\Rightarrow\mathrm{y}=\pm\frac{\mathrm{4}}{\:\sqrt{\mathrm{23}}} \\ $$$$\mathrm{Now}\:\mathrm{equation}\:\mathrm{of}\:\mathrm{tangent}\:\mathrm{line} \\ $$$$\mathrm{the}\:\mathrm{ellipse}\:\mathrm{is}\:\mathrm{5x}−\mathrm{y}=\frac{\mathrm{46}}{\:\sqrt{\mathrm{23}}}\:\mathrm{or} \\ $$$$\mathrm{5x}−\mathrm{y}=\mathrm{2}\sqrt{\mathrm{23}}\:,\:\mathrm{so}\:\mathrm{the}\:\mathrm{closest}\: \\ $$$$\mathrm{distance}\:\mathrm{is}\:\mathrm{required}\:\mathrm{equals}\:\mathrm{to}\: \\ $$$$\mathrm{d}\:=\:\frac{\mathrm{2}\sqrt{\mathrm{23}}}{\:\sqrt{\mathrm{26}}}\: \\ $$

Commented by liberty last updated on 23/Jan/21

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