Application of Derivatives

Find the point on the curve y 2 = 2x which is nearest to the point (1, 4). - Wired Faculty

Question

Find the point on the curve y² = 2x which is nearest to the point (1, 4).

Solution

Any point on the parabola y² = 2x is $straight P open parentheses 1 half straight t squared comma space space straight t close parentheses.$ Let Q be (1, 4)
Let a be the distance between $straight Q left parenthesis 1 comma space 4 right parenthesis space and space straight P open parentheses 1 half straight t squared comma space space straight t close parentheses$
$therefore space space space space straight d space equals space square root of open parentheses straight t squared over 2 minus 1 close parentheses squared plus left parenthesis straight t minus 4 right parenthesis squared end root space space space space space space space space space space space space space space space space space space rightwards double arrow space space space straight d squared space equals space open parentheses straight t squared over 2 minus 1 close parentheses squared plus left parenthesis straight t minus 4 right parenthesis squared therefore space space space space straight D space equals space straight t to the power of 4 over 4 minus straight t squared plus 1 plus straight t squared minus 8 straight t space plus 16 space equals space straight t to the power of 4 over 4 minus 8 straight t plus 17 comma space where space straight D space equals straight d squared$
Now, d is maximum or minimum when D is maximum or minimum.
$dD over dt space equals space straight t cubed minus 8$
$dD over dt space equals 0 space space space space space space space space space space space space space rightwards double arrow space space space space space space space straight t cubed minus 8 space equals space 0 space space space space space rightwards double arrow space space space straight t cubed space equals space 8 space space space space space rightwards double arrow space space space straight t space equals space 2 fraction numerator straight d squared straight D over denominator dt squared end fraction space equals space 3 straight t squared$
$At space straight t space equals space 2 comma space space space fraction numerator straight d squared straight D over denominator dt squared end fraction space equals space 3 left parenthesis 2 right parenthesis squared space equals 12 greater than 0 space$
$therefore space space space straight D space is space minimum space when space straight t space equals space 2 therefore space space space straight P space is space left parenthesis 2 comma space 2 right parenthesis comma space which space is space nearest space to space straight Q left parenthesis 1 comma space 4 right parenthesis$

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