Answer to Question #134444 in Analytic Geometry for Kimberly

Question #134444
Determine the equation of the line that has a distance 2 from the origin and is
parallel to the line that passes thru P1(2,7) and P2(-3,-2).
1
Expert's answer
2020-09-27T19:04:41-0400

calculate the vector

P1P2=(P2xP1x;P2yP1y)=(32;27)=(5;9)\overline{P_1P_2} = (P_{2x} - P_{1x};P_{2y} - P_{1y} ) = (-3-2;-2-7) = (-5;-9)

let ax + by + c =0 is equation of the line

then direction vector is (b;-a) and coincides with the vector P1P2\overline{P_1P_2}

hence b = -5

-a = -9

a = 9

9x- 5y + c = 0 equation of the line

h(ax+bx+c=0,(x0,y0))=ax0+by0+ca2+b2h(ax+bx+c=0,(x_0,y_0)) = \frac{|ax_0+by_0+c|} {\sqrt{a^2+b^2}}

where h is the distance from point to line

in our case h is equal to 2 and the coordinates of the point are equal to zero since they are the origin of coordinates

ax0+by0+ca2+b2=c92+52=c106\frac{|ax_0+by_0+c|} {\sqrt{a^2+b^2}} = \frac{|c|} {\sqrt{9^2+5^2}} = \frac{|c|} {\sqrt{106}}


hence

c=2106c = 2\sqrt{106} or c=2106c = -2\sqrt{106} and equation of the line

9x5y+2106=09x -5y + 2\sqrt{106} = 0

or 9x5y2106=09x -5y - 2\sqrt{106} = 0


Answer: 9x5y+2106=09x -5y + 2\sqrt{106} = 0 or 9x5y2106=09x -5y - 2\sqrt{106} = 0


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