locus question (1 Viewer)

ek102 · Aug 7, 2016

hi im a little confused,

Find the equation of the locus of a point that moves so that its distance from the line 3x+4y+5=0 is always 4 units.

InteGrand · Aug 7, 2016

ek102 said:
hi im a little confused,

Find the equation of the locus of a point that moves so that its distance from the line 3x+4y+5=0 is always 4 units.

$\noindent It'll be the pair of parallel lines parallel to the given line and shifted $4$ units away.$

$\noindent To get the equations of this pair of lines, let $P=(x,y)$. The condition for $P$ to be distance $4$ from the given line is that $\frac{\left|3x+4y+5\right|}{\sqrt{3^2 + 4^2}}=4$ (via perpendicular distance formula). This is equivalent to$

$\begin{align*}\frac{\left|3x+4y+5\right|}{5}&= 4 \\ \iff \left|3x+4y+5\right| &= 20 \\ \iff 3x+4y+5 = 20 &\text{ or }3x+4y+5 = -20 \\ \iff 3x+4y-15 = 0 &\text{ or }3x+4y+25 = 0 \quad (\text{these are the equations of the lines}). \end{align*}$

ek102 · Aug 8, 2016

thanks

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locus question (1 Viewer)

ek102

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InteGrand

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