Answer to Question #262424 in Analytic Geometry for Aliyah

Question #262424

A point moves so that the ratio of its distances from 3x + 4y + 8 = 0 and 4x + 3y – 6 = 0 is 2. Find the equation of its locus.


1
Expert's answer
2021-11-08T16:09:37-0500

A point moves so that the ratio of its distances from 3x+4y+8=03x + 4y + 8 = 0 and 4x+3y6=04x + 3y – 6 = 0 is 22. Let us find the equation of its locus. Let a point (x,y)(x,y) belongs to the locus. Then

3x+4y+832+42:4x+3y642+32=2.\frac{|3x + 4y + 8|}{\sqrt{3^2+4^2}}: \frac{|4x + 3y – 6|}{\sqrt{4^2+3^2}}=2.

It follows that

3x+4y+8=24x+3y6|3x + 4y + 8|=2|4x + 3y – 6|

which is equivalent to

3x+4y+82=44x+3y62.|3x + 4y + 8|^2=4|4x + 3y – 6|^2.

Then

9x2+16y2+64+24xy+48x+64y=4(16x2+9y2+36+24xy48x36y).9x^2+16y^2+64+24xy+48x+64y=4(16x^2+9y^2+36+24xy-48x-36y).

We conclude that

55x2+20y2+72xy144x80y+80=055x^2+20y^2+72xy-144x-80y+80=0

is the equation po its locus.


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