Application of Derivatives

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

Question

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

Solution

Any point on the parabola $straight y squared space equals 4 straight x space space is space straight P left parenthesis straight t squared comma space space 2 straight t right parenthesis.$
$Let space straight Q space be space left parenthesis 2 comma space 1 right parenthesis.$
Let d be the distance between Q (2, 1) and $straight P left parenthesis straight t squared comma space 2 straight t right parenthesis$
$therefore space space space space space space straight d space equals space square root of left parenthesis straight t squared minus 2 right parenthesis squared plus space left parenthesis 2 straight t minus 1 right parenthesis squared end root space space space rightwards double arrow space space space straight d squared space equals space left parenthesis straight t squared minus 2 right parenthesis squared space space plus space left parenthesis 2 straight t minus 1 right parenthesis squared therefore space space space space space straight D space equals space straight t to the power of 4 minus space 4 space straight t squared space plus space 4 space plus space 4 straight t squared space minus space 4 straight t space plus space 1 space space equals space straight t to the power of 4 minus 4 straight t plus 5 comma space space where space straight D space equals space straight d squared$
Now d is maximum or minimum when D is maximum or minimum.
$dD over dt space equals space 4 straight t cubed minus 4$
$dD over dt space equals space 0 space space rightwards double arrow space space space 4 straight t cubed minus 4 space equals space 0 space space space space rightwards double arrow space space straight t cubed minus 1 space equals space 0 space space space space rightwards double arrow space space straight t cubed space equals space 1 space space rightwards double arrow space space space straight t space equals space 1 fraction numerator straight d squared straight D over denominator dt squared end fraction space equals space 12 straight t squared$
$At space space space straight t space equals space 1 comma space space fraction numerator straight d squared straight D over denominator dt squared end fraction space equals space 12 left parenthesis 1 right parenthesis squared space equals space 12 space greater than 0 therefore space space space straight D space is space minimum space when space straight t space equals space 1 therefore space space space straight P space is space left parenthesis 1 comma space 2 right parenthesis comma space which space is space nearest space to space left parenthesis 2 comma space 1 right parenthesis$

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