MHB Distance between point and line

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To find the equation of the line through the origin that maintains a specific distance from points A(2,1) and B(1,4), the sum of the distances must equal \(\sqrt{8}\). The distance formula used is \[d=\frac{\left | Ax+By+C \right |}{\sqrt{A^{2}+B^{2}}}\], leading to the equation \[\frac{-2A-B}{\sqrt{A^{2}+B^{2}}}+\frac{A+4B}{\sqrt{A^{2}+B^{2}}}=\sqrt{8}\]. By recognizing that the line passes through the origin, the constant C is set to 0, simplifying the equation to involve only one variable, A. This approach allows for the determination of the line's slope as \(-A\).
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Hello all,

I need your assistance with the following:

Find the equation of the line that goes through the origin (0,0) if it is known that the sum of distances of the line from A(2,1) and B(1,4) is equal to \[\sqrt{8}\], and it is also known that the line is between the points.

The formula I am supposed to use is:

\[d=\frac{\left | Ax+By+C \right |}{\sqrt{A^{2}+B^{2}}}\]

What I did, is I set the equation to be:

\[\frac{-2A-B-C}{\sqrt{A^{2}+B^{2}}}+\frac{A+4B+C}{\sqrt{A^{2}+B^{2}}}=\sqrt{8}\]

The absolute value was omitted since the line is between the points, i.e. one point is below and one above.

I am stuck with 1 equation and 2 variables. How do I proceed ? Since the point (0,0) is on the line, I know that C is 0 in the equation Ax+By+C=0, but this parameter is not important anyway (C-C=0).

Thank you !
 
Mathematics news on Phys.org
The family of lines passing through the origin may be written:

$$Ax+y=0$$

where the slope is $-A$. Now you will have only 1 variable in your equation. ;)
 
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