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Find the equations of the locus of the point $P(x,y)$ that is equidistant from the lines $4x+3y-2=0$ and $12x-5y+6=0$.

Do I use simultaneous equations? I cant remember :(

Ben Longo
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Kate
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  • What's the formula for the distance from a point to a line? – Ben Longo Nov 14 '15 at 05:14
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    This is the same as finding the angle bisector of the two lines. See http://math.stackexchange.com/questions/38665/equation-of-angle-bisector-given-the-equations-of-two-lines-in-2d – Nicholas Nov 14 '15 at 05:40

1 Answers1

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The distance $D$ from a point $(x_0, y_0)$ to a line $ax+by+c=0$ is given by

$$D=\frac{\vert ax_0+by_0+c \vert}{\sqrt{a^2+b^2}}$$

For the point $P(x,y)=(x_0, y_0)$ to be equidistant from the 2 lines $4x+3y-2=0$ and $12x-5y+6=0$,

$$\frac{\vert 4x_0+3y_0-2 \vert}{5}=\frac{\vert 12x_0-5_y+6 \vert}{13}$$

Ben Longo
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