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Question

The coordinates of a point on the hyperbola x224y218=1 which is nearest to the line 3x+2y+1=0 are?

A
(6,3)
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B
(6,3)
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C
(6,3)
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D
(5,3)
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Solution

The correct option is D (6,3)
The shortest distance would be perpendicular to the line
3x+2y+1=0
2y=3x1
y=(32)x12
A point on hyperbola that had same slope could be at minimum distance.
Slope of hyperbola by differentiating
x224y218=1
124(2xdx)(118)(2ydy)=0
xdx12=ydy9
dxdy=3x4y=32
6x=12y
x=2y
Substituting in given hyperbola equation -
(2y)22yy218=1
y26y218=1
(318).y2(918)y2=1
(218)y2=1
y2=9
y=+3 and y=3
Then, x224918=1
x224=32
x2=36
x=6 and x=6
so, point are (6,3) and (6,3)
perpendicular line that would be shortest distance to original line =23
we have slope and point so the line ,
y3=23[x(6)]
y3=23(x+6)
y=23x+7.
Hence, solved.



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