SOLUTION: find an equation of a circle that tangent is to 3x+4y-15=0 at (5,0); contains (-2,-1)

Question 669435: find an equation of a circle that tangent is to 3x+4y-15=0 at (5,0); contains (-2,-1)
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
Maybe there is a formula in your book that solves this type of problem.
I do not do types or formulas, so my solution will just stem from the most basic concepts. Sorry if it gets long and boring.

CIRCLE TANGENT TO A LINE:
At the point where a circle is tangent to a line, the radius is perpendicular to that line.
So the center of that circle is on the line perpendicular to at point (5,0).
<--> <--> <--> ,
so the line has a slope of
The slope of a line perpendicular to is
=
The equation of a line with slope that passes through (5,0) is
<-->

CIRCLE PASSING THROUGH TWO POINTS:
The center of the circle that passes through (5,0) and (-2,-1) is at the same distance from those two points,
so it is on the perpendicular bisector of the segment joining those two points.
The perpendicular bisector is perpendicular to that segment and passes through the midpoint of that segment.
The midpoint of that segment, calculated from the coordinates of (5,0) and (-2,-1) is
(,)=(,)
The slope of that segment, calculated from the coordinates of (5,0) and (-2,-1) is

and the slope of a perpendicular line is =
The equation of the line with slope passing through (,) is
<--> <--> <--> <-->

THE CENTER OF OUR CIRCLE:
The center of the circle we are looking for is on the lines
and .
Those lines are not parallel, so they intersect at one point.
That point is the center of the circle.
We need to solve
--> (multiplying both sides times 3)
--> --> --> --> --> -->
Substituting into ,
--> -->
So the center of the circle is (2,-4).

THE RADIUS OF THE CIRCLE:
The radius is the distance from circle center (2,-4) to circle point (5,0).
We can calculate

THE EQUATION OF THE CIRCLE"
The equation of a circle centered at (2,-4) and with is
-->
RELATED QUESTIONS
tangent to 3x+4y-15=0 at (5,0) contains (0,4) find the equation of a... (answered by Alan3354,keyaytee)

Find an equation of a circle tangent to 3x+4y-16=0 at (4,1) with a radius of... (answered by Alan3354)

Find an equation(s) of the circle(s) tangent to y = 0 at (4,0); tangent to 3x - 4y - 17 = (answered by Edwin McCravy)

Equation of a circle Passing through (6,-1) and tangent 4y-3x-2=0 at... (answered by Alan3354,ikleyn)

find the equation of a circle tangent to 3x+4y-2=0 with center... (answered by Edwin McCravy)

find equation of the circle that has radius of 5 and tangent to the line 3x+4y=24 at the... (answered by josgarithmetic)

Find the equation of the circle with C(-4,2) an is the tangent to the linne... (answered by robertb)

What is the equation of the circle with center at (0, 2) and tangent to the line 3x - 4y... (answered by ikleyn)

What is the equation of the circle with center at(0.2) and tangent to the line 3x-4y=12 (answered by Alan3354,ikleyn)