what is the equation of a line parallel to a given line say y=x at a constant disance of 1 unit from it? I guess there will be 2 equations,one above x axis and other below x axis
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Equation of any line parallel to $ y=x $ is $ y=x+k $. Find $ k $ according to your requirement. Yes there will be two lines but not the way you described. – Debashish Jun 09 '15 at 10:23
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how to find k? using the information that it is at a distance of 1 unit from y=x? – user3366497 Jun 09 '15 at 10:26
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You can use the formula for distance of a point $(x_1,y_1)$ from the line $ax+by+c=0$, which is given by $\frac{|ax_1+by_1+c|}{\sqrt{a^2+b^2}}$ – Debashish Jun 09 '15 at 10:27
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The required lines are not $y=x+1$ and $y=x-1$ .. !! – Debashish Jun 09 '15 at 10:28
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how can they be y=x+1 or x-1!! not possible – user3366497 Jun 09 '15 at 10:29
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yes they are not the answer ...find $k$ using the above formula – Debashish Jun 09 '15 at 10:29
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1will the answer be y=x+root(2) and y=x-root(2)? – user3366497 Jun 09 '15 at 10:32
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yes, it's correct – Debashish Jun 09 '15 at 10:34
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The distance of a point $P=(x_P,y_P)$ from a stright line of equation $ax+by+c=0$ is given by: $$ d=\dfrac{|ax_P+by_P+c|}{\sqrt{a^2+b^2}} $$ (see here)
so, a generic point $(x,y)$ at distance $1$ from the line is such that: $$ \dfrac{|ax+by+c|}{\sqrt{a^2+b^2}}=1 $$ and , separating the two case for the absolute value, you have the equations of two stright lines.
Emilio Novati
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