CAT Exam Model Paper 2 with solutions for free online practice

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Question : 96 of 100

Marks: +1, -0

Find the shortest distance form the point (0, 8a) to the curve a x

^{2}

= y

^{3 }

2a units
2a $√11$ units
2 $√11$ units
$a /2$ units

Solution:

( b )

d = \√ { ( x - 0 ) ^ { 2 } + ( y - 8 a ) ^ { 2 } }

d = √{ x ^ { 2 } + ( y - 8 a ) ^ { 2 } }

From

a x ^ { 2 } = y ^ { 3 } , x ^ { 2 } = { 1 } /{ a } y ^ { 3 }

d = √ { { 1 }/ { a } y ^ { 3 } + ( y - 8 a ) ^ { 2 } }

{ d d } /{ d y } = { { 3 }/ { a } y ^ { 2 } + 2 ( y - 8 a ) } /{ 2 √ { { 1 }/ { a } y ^ { 3 } + ( y - 8 a ) ^ { 2 } } }

{ 3 }/ { a } y ^ { 2 } + 2 y - 16 a = 0

3 y ^ { 2 } + 2 a y - 16 a ^ { 2 } = 0

y = { - 2 a ±√{ 4 a ^ { 2 } - 4 ( 3 ) ( - 16 a ^ { 2 } ) } } /{ 2 ( 3 ) }

y = { - 2 a ± 14 a } /{ 6 }

y = 2 a

and

- { 8 }/ { 3 } a

y = - { 8 }/ { 3 } a

is meaning less, use

y = 2 a

d = √ { { 1 }/ { a } ( 2 a ) ^ { 3 } + ( 2 a - 8 a ) ^ { 2 } }

d = √{ { 1 } /{ a } ( 2 a ) ^ { 3 } + ( 2 a - 8 a ) ^ { 2 } } , d = √{ 44 a ^ { 2 } }

d = 2 a √ { 11 }

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