Let $P (x_{1}, y_{1})$ $P(x_1, y_1)$ be a point and let $l$ $l$ be the line with equation $a x + b y + c = 0$ $ax+ by +c =0$ . Show the distance $d$ $d$ from $P \to l$ $P->l$ is given by: $d = \frac{a x_{1} + b y_{1} + c}{\sqrt{a^{2} + b^{2}}}$ $d =(ax_1+ by_1 + c)/sqrt( a^2 +b^2)$ ? Find the distance $d$ $d$ of the point P(6,7) from the line l with equation 3x +4y =11?

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Let $P (x_{1}, y_{1})$ $P(x_1, y_1)$ be a point and let $l$ $l$ be the line with equation $a x + b y + c = 0$ $ax+ by +c =0$ . Show the distance $d$ $d$ from $P \to l$ $P->l$ is given by: $d = \frac{a x_{1} + b y_{1} + c}{\sqrt{a^{2} + b^{2}}}$ $d =(ax_1+ by_1 + c)/sqrt( a^2 +b^2)$ ? Find the distance $d$ $d$ of the point P(6,7) from the line l with equation 3x +4y =11?

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Answer 1 · 2016-11-16T00:54:55

$d = 7$ $d = 7$

Explanation:

Let $l \to a x + b y + c = 0$ $l->a x + b y + c=0$ and $p_{1} = (x_{1}, y_{1})$ $p_1 = (x_1,y_1)$ a point not on $l$ $l$ .

Supposing that $b \neq 0$ $b ne 0$ and calling $d^{2} = {(x - x_{1})}_{1}^{2} + {(y - y_{1})}_{1}^{2}$ $d^2=(x-x_1)^2+(y-y_1)^2$ after substituting $y = - \frac{a x + c}{b}$ $y=-(a x+c)/b$ into $d^{2}$ $d^2$ we have

$d^{2} = {(x - x_{1})}_{1}^{2} + {(\frac{c + a x}{b} + y_{1})}_{1}^{2}$ $d^2=(x - x_1)^2 + ((c + a x)/b + y_1)^2$ . The next step is find the $d^{2}$ $d^2$ minimum regarding $x$ $x$ so we will find $x$ $x$ such that

$\frac{d}{d x} (d^{2}) = 2 (x - x_{1}) - \frac{2 a (\frac{c + a x}{b} + y_{1})}{b} = 0$ $d/(dx)(d^2) = 2 (x - x_1) - (2 a ((c + a x)/b + y_1))/b = 0$ . This occours for

$x = \frac{b^{2} x_{1} - a b y_{1} - a c}{a^{2} + b^{2}}$ $x = (b^2 x_1 - a b y_1-a c)/(a^2 + b^2)$ Now, substituting this value into $d^{2}$ $d^2$ we obtain

$d^{2} = \frac{{(c + a x_{1} + b y_{1})}_{1}^{2}}{a^{2} + b^{2}}$ $d^2=(c + a x_1 + b y_1)^2/(a^2 + b^2)$ so

$d = \frac{c + a x_{1} + b y_{1}}{\sqrt{a^{2} + b^{2}}}$ $d = (c + a x_1 + b y_1)/sqrt(a^2 + b^2)$

Now given

$l \to 3 x + 4 y - 11 = 0$ $l->3x+4y-11=0$ and $p_{1} = (6, 7)$ $p_1=(6,7)$ then

$d = \frac{- 11 + 3 \times 6 + 4 \times 7}{\sqrt{3^{2} + 4^{2}}} = 7$ $d = (-11+3xx6+4xx7)/sqrt(3^2+4^2)=7$

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Let P(x1,y1)P(x_1, y_1) be a point and let ll be the line with equation ax+by+c=0ax+ by +c =0. Show the distance dd from P→lP->l is given by: d=ax1+by1+c√a2+b2d =(ax_1+ by_1 + c)/sqrt( a^2 +b^2)? Find the distance dd of the point P(6,7) from the line l with equation 3x +4y =11?

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