Author Topic: Locus (Read 2919 times) Tweet Share

Nialllovespie · « **on:** October 11, 2016, 05:10:34 pm »

How do I solve this problem:

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.

The answer is 3x+4y-15=0, 3x+4y+25=0 (if that's any help) but I have no idea how to solve it

.

Thank you so much for your help!!

RuiAce · « **Reply #1 on:** October 11, 2016, 05:31:53 pm »

Quote from: Nialllovespie on October 11, 2016, 05:10:34 pm

How do I solve this problem:

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.

The answer is 3x+4y-15=0, 3x+4y+25=0 (if that's any help) but I have no idea how to solve it.

Thank you so much for your help!!

$\text{When we talk about distance from a line}\\ \text{We are always interested in the }\textit{perpendicular distance}\\ \text{of that line to the point.}$
$\text{One way to approach it is to hence, throw it straight into the perpendicular distance formula.}\\ \text{Let the point }P\text{ be }(x,y).\text{ Then}\\ \begin{align*}PA&=4\\\frac{|3x+4y+5|}{\sqrt{3^2+4^2}}&=4\\ |3x+4y+5|&=20\end{align*}$
$\text{So by splitting it up:}\\ 3x+4y+5=-20 \iff 3x+4y+25=0\\ \text{or}\\ 3x+4y+5=20 \iff 3x+4y-15=0$
________________________
$\text{Observe that this is much more difficult than the classic}\\ \textit{A point }P\textit{ moves so that its always }2\textit{ units from }(4,2)\\ \text{Where you would start with }PA=2, PA^2=4\text{ but use the }\textbf{distance}\text{ formula.}$

pels · « **Reply #2 on:** October 19, 2016, 01:22:32 pm »

So when do you know when to use distance formula or perpendicular distance formula for locus questions

Also, for the one you mentioned:

P moves so that it is 2 units away from (4,2), is it just a circle with centre (4,2) radius 2?

Is that correct?

RuiAce · « **Reply #3 on:** October 19, 2016, 01:24:33 pm »

Yeah that easy one becomes a circle.

You always tend to the distance formula if you can. The perpendicular distance formula should NEVER be the number 1 option; it is an alternative.

Note that perpendicular distance demands that a line be present. Without a line, you can't even use it.

pels · « **Reply #4 on:** October 19, 2016, 01:31:26 pm »

Yeah thought so.

Cheers Rui

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Author Topic: Locus (Read 2919 times) Tweet Share

Nialllovespie

Locus

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pels

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RuiAce

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pels

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