Need help with locus question? (1 Viewer)

MysteryMoon · May 21, 2015

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

The answer is 3x + 4y + 25 = 0, -3x + 4y - 15 = 0

Thanks.

InteGrand · May 21, 2015

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

The answer is 3x + 4y + 25 = 0, -3x + 4y - 15 = 0

Thanks.

$Let the point by $P(x,y)$. By the perpendicular distance formula, the distance of $P$ to the line $3x + 4y + 5 = 0$ (which is 4 units) is given by$

$4=\text{dist.}=\frac{|3x + 4y + 5|}{\sqrt{3^2 + 4^2}}$

$\Rightarrow 4 = \frac{|3x + 4y + 5|}{5}$

$\Rightarrow 20 = |3x + 4y + 5|$

$\Rightarrow 20 = 3x + 4y + 5$ or $-20 = 3x + 4y + 5$

$$\Rightarrow 3x+4y-15=0$ or $3x + 4y + 25=0$, which is clearly a pair of parallel lines (parallel to the line $3x + 4y + 5 =0$ and each one shifted out 4 units from that line in the direction perpendicular to that line).$

MysteryMoon · May 21, 2015

Thanks for your help! ^_^

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