SOLUTION: The circle is tangent to the line 4x+3y=4 at the point (4,-4) and center is on the line x-y=7. Find the equation of the circle. (I'm struggling to find the center in x-y=7 that say

Algebra ->  Circles -> SOLUTION: The circle is tangent to the line 4x+3y=4 at the point (4,-4) and center is on the line x-y=7. Find the equation of the circle. (I'm struggling to find the center in x-y=7 that say      Log On


   

Question 1139514: The circle is tangent to the line 4x+3y=4 at the point (4,-4) and center is on the line x-y=7. Find the equation of the circle. (I'm struggling to find the center in x-y=7 that says to be perpendicular to the center itself and to the point of tangency)
Answer by greenestamps(13206) About Me  (Show Source):
You can put this solution on YOUR website!


Put the equations in slope-intercept form.

4x%2B3y=4 --> y+=+-%284%2F3%29x%2B4%2F3
x-y=7 --> y+=+x-7

If the circle is tangent to the line y+=+-%284%2F3%29x%2B4%2F3 at (4,-4), then the center of the circle has to lie on the line with slope 3/4 passing through (4,-4). (A radius to a point of tangency is perpendicular to the tangent; slopes of perpendicular lines are negative reciprocals.)

y+=+%283%2F4%29x%2Bb
-4+=+%283%2F4%294%2Bb
-4+=+3%2Bb
b+=+-7

So the center of the circle lies on the line y+=+%283%2F4%29x-7.

The given information says that the center of the circle lies on the line y+=+x-7.

So the center of the circle is at the one point that lies on both y+=+%283%2F4%29x-7 and y+=+x-7.

Solve that pair of equations to find the intersection point; that is the center of the circle.