SOLUTION: WHAT IS THE EQUATION OF A CIRCLE TOUCHING THE LINES {{{ x - 3y - 11 = 0 }}} AND {{{ 3x - y - 9 = 0 }}} AND, HAVING ITS CENTER ON THE LINE {{{ x + 2y + 19 = 0 }}} ?

Algebra ->  Circles -> SOLUTION: WHAT IS THE EQUATION OF A CIRCLE TOUCHING THE LINES {{{ x - 3y - 11 = 0 }}} AND {{{ 3x - y - 9 = 0 }}} AND, HAVING ITS CENTER ON THE LINE {{{ x + 2y + 19 = 0 }}} ?      Log On


   

Question 877020: WHAT IS THE EQUATION OF A CIRCLE TOUCHING THE LINES +x+-+3y+-+11+=+0+ AND +3x+-+y+-+9+=+0+ AND, HAVING ITS CENTER ON THE LINE +x+%2B+2y+%2B+19+=+0+ ?
Answer by josgarithmetic(39630) About Me  (Show Source):
You can put this solution on YOUR website!
Take a viewpoint from above the plane containing the lines, with the three lines graphed.

graph%28300%2C300%2C-12%2C12%2C-12%2C12%2Cx%2F3-11%2F3%2C3x-9%2C-x%2F2-19%2F2%29
The blue line is the one which contains the center of the circle.

This will be messy, but you might use Distance formula. The general point of the center of circle is (x, -x/2-19/2). General point of one of the lines is (x, x/3-11/3) and of the other line is (x, 3x-9).

The center point is equadistant from the two other general points.

This is what becomes messy:

That should be done on paper and solve for x. This will let you find y, and these are coordinates for the center of the circle.

The distance formula equation shown above seems to simplify to 416x%5E2-224x-1216=0. Further reduces through factorizations GCF of 2%5E5, to
highlight_green%2813x%5E2-7x-38=0%29.