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# Mathematics - Locus of a Point

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Now I have 3 answers to this question non correct. One tutor has posted conflicting solutions.

Find the equation(s) of the locus of P that moves such that the distance from P to the lines 3x -4y + 1 = 0 and 12x + 5y + 3 = 0 is in the ratio 3:1

Nov 24th, 2017

I am glad I found you again!

Sorry dude, my answer cut off in the middle and, so, I gave you only half of the answer first time. Just realized it! That’s why it looks so weird at the end first time!

3x - 4y + 1 = 0

12x + 5y + 3 = 0

Simplify, Find x from the first equation and substitute x with the final result in a second equation to find y within the next steps:

1)

3x = 4y - 1,  x = (4 y -1) / 3

5y = - 3 - 12x,   y = - (3 + 12x) / 5

2)

x = (4 (-(3+12x)/5 ) - 1) / 3

x = (((- 12 - 48x) / 5) - 1) /3

x = (( -12 - 48x - 5) / 5) /3

x = ((- 17 - 48x)/5 ) /3

x = -17/15 - 48x/15 = (-17 - 48x)/15

x - ((17+48x)/15) = 0

x/15 - (17+48x)/15 = 0

(15x + 48x + 17)/15 = 0

(63x + 17)/15 =0

63x = -17

x = - 17/63

3)

y = -( (3 + 12(- 17/63))) /5

y = - (( 3 – 204/63) )/5

y = - ((189 - 204)/63)/5

y = (15/63)/5

y = 1/21

4)

D= │ax + by + c│ / √a 2 + b2

You can take any of those condition equations in your question. Let’s take first:

3x - 4y + 1 = 0

Here, a = 3; b=4; c = 1

Substitute and plug them into equation for D in a ration 1:3 as stated:

1/3 = │3x17/63 + 4x1/21 + C│/ 5

5/3 = │17/21 + 4/21 + C│

5/3 = 21/21 + C

5/3 – C = +- 1

C1 = 5/3 + 1 = 8/3

C2 = 5/3 – 1 = 2/3

2 is 3 times smaller than 8, corresponding to the 1:3 ratio

Having C1 and C2 in chosen equation, you will end up with two equations as:

1. 3x - 4y + 8/3 = 0

2. 3x - 4y + 2/3 = 0

Hope this answer found and got you on time!

Dec 19th, 2014

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Nov 24th, 2017
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