Q. 63 f(x,y)=yx,P=(1,9)... [FREE SOLUTION]

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Chapter 12: Q. 63 (page 954)

$f (x, y) = \sqrt{\frac{y}{x}}, P = (1, 9)$

Short Answer

Expert verified

The equation's solution $f (x, y) = \sqrt{\frac{y}{x}}, P = (1, 9)$ is $9 x 鈭� y + 6 z = 18$

Step by step solution

Step 1:Given data

$f (x, y) = \sqrt{\frac{y}{x}} at point P = (x_{0}, y_{0}) = (1, 9)$

The line of tangent equation is

$f_{x} (x_{0}, y_{0}) (x 鈭� x_{0}) + f_{y} (x_{0}, y_{0}) (y 鈭� y_{0}) = z 鈭� f (x_{0}, y_{0})$

$f_{x} (1, 9) (x 鈭� 1) + f_{y} (1, 9) (y 鈭� 9) = z 鈭� f (1, 9)$ $(1)$

Step 2:Find fx

$f_{x} (x_{0}, y_{0}) = \frac{d}{d x} (\sqrt{\frac{y}{x}}) \frac{d}{d x} {(\frac{y}{x})}_{(x_{0}, y_{0})}$

$f_{x} (x_{0}, y_{0}) = {\frac{1}{2 \sqrt{\frac{y}{x}}} (鈭� \frac{y}{x^{2}})|}_{(x_{0}, y_{0})}$

$f_{x} (1, 9) = {\frac{1}{2 \sqrt{\frac{y}{x}}} (鈭� \frac{y}{x^{2}})|}_{(1, 9)}$

$f_{x} (1, 9) = \frac{1}{2 \sqrt{\frac{9}{1}}} (鈭� \frac{9}{1^{2}})$

$f_{x} (1, 9) = 鈭� \frac{9}{6}$ $(2)$

Step 3:Find fy

$f_{y} (x_{0}, y_{0}) = {\frac{d}{d y} f (x, y)|}_{(x_{0}, y_{0})} = {\frac{d}{d y} \sqrt{\frac{y}{x}}|}_{(x_{0}, y_{0})}$

$f_{y} (x_{0}, y_{0}) = {\frac{d}{d y} (\sqrt{\frac{y}{x}}) \frac{d}{d y} (\frac{y}{x})|}_{(x_{0}, y_{0})}$ $f_{y} (x_{0}, y_{0}) = {\frac{1}{2 \sqrt{\frac{y}{x}}} 鈰� \frac{1}{x}|}_{(x_{0}, y_{0})}$

localid="1650515198116" $f_{y} (1, 9) = \frac{1}{{2 x \sqrt{\frac{y}{x}}|}_{(1, 9)}}$

localid="1650515201090" $f_{y} (1, 9) = \frac{1}{6}$ localid="1650515204300" $(3)$

Findf(1,9)

$f (x_{0}, y_{0}) = f (1, 9) = \sqrt{\frac{9}{1}}$

$f (1, 9) = 3$ $(4)$

Step 5:Solution

Substituting equation $f_{x} (1, 9) = 鈭� \frac{9}{6}$ , $f_{y} (1, 9) = \frac{1}{6}$ and $f (1, 9) = 3$ in $f_{x} (1, 9) (x 鈭� 1) + f_{y} (1, 9) (y 鈭� 9) = z 鈭� f (1, 9)$ get

$鈭� \frac{9}{6} (x 鈭� 1) + \frac{1}{6} (y 鈭� 9) = z 鈭� 3$

$鈭� \frac{9}{6} x + \frac{9}{6} + \frac{1}{6} y 鈭� \frac{9}{6} = z 鈭� 3$

$鈭� \frac{9}{6} x + \frac{1}{6} y 鈭� z = 鈭� 3$

Multiple $6$ on both side

$\begin{aligned} 鈭� 9 x + y 鈭� 6 z = 鈭� 18 \\ 9 x 鈭� y + 6 z = 18 \end{aligned}$

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91影视

$f (x, y) = \sqrt{\frac{y}{x}}, P = (1, 9)$

Short Answer

Step by step solution

Step 1:Given data

Step 2:Find fx

Step 3:Find fy

Findf(1,9)

Step 5:Solution

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