Find the point on the hyperbola xy = 8 that is closest to the point (3,0)

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Louis Smith · Accepted Answer

Given xy=8so,y=8xnow any point of given hyperbola can be written as (x,8x)so,Let&#039;s that point on hyperbola is (x,8x)Now we have to find distance between (x,8x) and (3,0), and minimize the distancedistance between two given points (x1,y1) and (x2,y2) are given byd=(x2−x1)2+(y2−y1)2so,d=(x−3)2+(8x−0)2d=(x−3)2−64x2for finding minimum value of distance we differentiate d with respect to x and equate with zero, and find value of x where d&#039; is zeroso, first squaring both side then differentiatingd2=(x−3)2+64x22dd′=2(x−3)−64x3=02(x−3)X3−64x3=0=2(x−3)x3−64x3=0=2[x3(x−3)−32}{x3}=0=x4−3x3−32=0at x=4,x4−3x3−32=0 becomes zeroso, that point will be (4,84)≡(4,2)hence, point on hyperbola whose distance from (3,0) will be minimum is (4,2).

Find the point on the hyperbola xy = 8 that

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