Conic Sections

Hey..I don't understand conic sections at all...

1.
Find a polynomial equation with integer coeffiecients for the set of coplanar points desscribed. Tell whether or not the graph is a conic section and, if it is, tell which conic section.

For each point, its distance from the fixed point (4,3) is 3 times its disstance from the fixed point (-1,2).

Hello, Trentt!

Quote:

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1. Find a polynomial equation with integer coeffiecients
for the set of coplanar points described.
Tell whether or not the graph is a conic section and, if it is, tell which conic section.

For each point, its distance from the fixed point A:(4,3)
is 3 times its distance from the fixed point B:(-1,2).

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Let the variable point be P:(x,y).
Then: .PA .= .3·PB
. . . . . . . . ._______________ . . . . . . _______________
We have: .√(x - 4)² + (y - 3)² . = . 3·√(x + 1)² + (y - 2)²

Square both sides: .x² - 8x + 16 + y² - 6y + 9 .= .9x² + 18x + 9 + 9y² - 36y + 36

And we have: .8x² + 8y² + 26x - 30y + 20 .= .0 . → . 4x² + 4y² + 13x - 15y + 10 .= .0

Since the coefficients of x² and y² are equal, the conic is a circle.
. . In fact, it is called a Circle of Apollonius.