Find the point on the hyperbola xy = 8 that

TokNeekCepTdh 2021-11-15 Answered
Find the point on the hyperbola xy = 8 that is closest to the point (3,0)
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Expert Answer

Louis Smith
Answered 2021-11-16 Author has 14 answers
Given xy=8
so,
y=8x
now any point of given hyperbola can be written as (x,8x)
so,
Let's that point on hyperbola is (x,8x)
Now we have to find distance between (x,8x) and (3,0), and minimize the distance
distance between two given points (x1,y1) and (x2,y2) are given by
d=(x2x1)2+(y2y1)2
so,
d=(x3)2+(8x0)2
d=(x3)264x2
for finding minimum value of distance we differentiate d with respect to x and equate with zero, and find value of x where d' is zero
so, first squaring both side then differentiating
d2=(x3)2+64x2
2dd=2(x3)64x3=0
2(x3)X364x3=0
=2(x3)x364x3=0
=2[x3(x3)32}{x3}=0
=x43x332=0
at x=4,x43x332=0 becomes zero
so, that point will be (4,84)(4,2)
hence, point on hyperbola whose distance from (3,0) will be minimum is (4,2).
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