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Application Of Derivatives

Question
CBSEENMA12035478
Wired Faculty App

The point on the curve x2 = 2y which is nearest to the point (0, 5) is

  • left parenthesis 2 square root of 2 comma space space 4 right parenthesis
  • left parenthesis 2 square root of 2 comma space space 0 right parenthesis

  • (0, 0)

  • (2, 2)

Solution

A.

left parenthesis 2 square root of 2 comma space space 4 right parenthesis

Let (x, y) be the point on straight x squared space equals space 2 straight y which is nearest to the point (0, 5).
because space space space space left parenthesis straight x comma space straight y right parenthesis space lies space on space the space curve space straight y space equals space straight x squared over 2 therefore space space space space space space point space is space open parentheses straight x comma space space straight x squared over 2 close parentheses Let space straight d space be space the space distance space between space left parenthesis 0 comma space 5 right parenthesis space and space open parentheses straight x comma space straight x squared over 2 close parentheses. therefore space space space space space straight d space equals space square root of left parenthesis straight x minus 0 right parenthesis squared plus open parentheses straight x squared over 2 minus 5 close parentheses end root space space space space rightwards double arrow space space space space space straight d squared space equals straight x squared plus open parentheses straight x squared over 2 minus 5 close parentheses squared Put space straight d squared space equals space straight D. space space space space space space therefore space space space space space straight D space equals space straight x squared plus space open parentheses straight x squared over 2 minus 5 close parentheses squared
                              dD over dx space equals space 2 straight x plus 2 open parentheses straight x squared over 2 minus 5 close parentheses. space fraction numerator 2 straight x over denominator 2 end fraction space equals 2 straight x plus straight x left parenthesis straight x squared minus 10 right parenthesis
                                        equals 2 straight x plus straight x cubed minus 10 straight x equals space space straight x cubed minus 8 straight x
                    dD over dx space equals 0 space space space give space us space straight x cubed minus 8 straight x space equals space 0
therefore space space space space straight x left parenthesis straight x squared minus 8 right parenthesis space equals space 0 space space space space space rightwards double arrow space space space space straight x space equals space 0 comma space space space plus-or-minus square root of 8 space equals space 0 comma space plus-or-minus 2 square root of 2 space space space space space space space space space space fraction numerator straight d squared straight D over denominator dx squared end fraction space equals space 3 straight x squared minus 8 At space space straight x space equals space 0 comma space space space fraction numerator straight d squared straight D over denominator dx squared end fraction space equals space 0 minus 8 space equals space space minus 8 space less than 0 space therefore space space space space space straight D space is space maximum space when space straight x space equals space 0
We are not interested in this case.
At space space straight x space equals space 2 square root of 2 comma space space space fraction numerator straight d squared straight D over denominator dx squared end fraction space equals space 3 space cross times space 8 space minus space 8 space space equals space 16 space greater than space 0 therefore space space space straight D space is space minimum space when space straight x space equals space 2 square root of 2 comma space space space straight y space equals space 8 over 2 space equals space 4 therefore space space space point space is space left parenthesis 2 square root of 2 comma space space 4 right parenthesis
At space space straight x space equals space minus 2 square root of 2 comma space space space space space fraction numerator straight d squared straight D over denominator dx squared end fraction space equals space 3 space cross times 8 space minus space 8 space equals space 16 space greater than space 0 space therefore space space space space straight D space is space minimum space when space straight x space equals space minus 2 square root of 2 comma space space space straight y space equals space 8 over 2 space equals space 4 therefore space space space space point space is space left parenthesis negative 2 square root of 2 comma space space 4 right parenthesis therefore space space space left parenthesis straight A right parenthesis space is space correct space answer.

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

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