Help with locus question? (1 Viewer)

MysteryMoon · May 21, 2015

Find the equation of the locus of a point that moves so that it is equidistant from the line 4x - 3y + 2= 0 and the line 3x + 4y -7 = 0

The answers are: x - 7y + 9 = 0, 7x + y -5 =0

InteGrand · May 21, 2015

MysteryMoon said:
Find the equation of the locus of a point that moves so that it is equidistant from the line 4x - 3y + 2= 0 and the line 3x + 4y -7 = 0

The answers are: x - 7y + 9 = 0, 7x + y -5 =0

$Let the point be $P(x,y)$. By the perpendicular distance formula, the distance of $P$ to the lines $4x -3y + 2 = 0$ and $3x+4y-7=0$ (which is alway equal) is given by$

$\frac{|3x +4y -7| }{\sqrt{3^2 + 4^2}}=\text{dist.}=\frac{|4x -3y + 2|}{\sqrt{4^2 + (-3)^2}}$

$\Rightarrow \frac{|3x +4y -7| }{\sqrt{3^2 + 4^2}} = \frac{|4x -3y + 2|}{\sqrt{4^2 + (-3)^2}}$

$$\Rightarrow |3x +4y -7| = |4x -3y + 2|$ (the denominators from either side cancel as they are equal)$

$\Rightarrow 3x +4y -7= 4x -3y + 2$ or $3x +4y -7 = -(4x -3y + 2)$

$$\Rightarrow x-7y+9=0$ or $7x +y -5=0$.$

MysteryMoon · May 21, 2015

Thanks so much!!!

Help with locus question? (1 Viewer)

MysteryMoon

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InteGrand

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MysteryMoon

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