SOLUTION: Find an equation of a circle tangent to 3x+4y-16=0 at (4,1) with a radius of 5.

Question 783735: Find an equation of a circle tangent to 3x+4y-16=0 at (4,1) with a radius of 5.
Answer by Alan3354(69443) (Show Source): You can put this solution on YOUR website!
Find an equation of a circle tangent to 3x+4y-16=0 at (4,1) with a radius of 5.
----------------
The line thru the center of the circle is perpendicular to the given line, and the center is 5 units from the tangent points.
---
Find the slope of the line
3x+ 4y = 16
solve for y
y = -3x/4 + 4
Slope of the line = -3/4
Slope of the perpendicular line = 4/3
---
Find the eqn of the perpendicular line thru (4,1)
use y = mx + b and the point to find b, the y-intercept
1 = (4/3)*4 + b
b = -13/3
-----
The perpendicular line is y = 4x/3 - 13/3
---
There are 2 points 5 units from (4,1) on y = 4x/3 - 13/3 --> 2 tangent circles
-----
Draw a circle radius = 5 centered at (4,1)

y = 4x/3 - 13/3
Sub for y

Solve for x

Solved by pluggable solver: SOLVE quadratic equation (work shown, graph etc)

Quadratic equation (in our case ) has the following solutons:

For these solutions to exist, the discriminant should not be a negative number.

First, we need to compute the discriminant : .

Discriminant d=22500 is greater than zero. That means that there are two solutions: .

Quadratic expression can be factored:

Again, the answer is: 7, 1. Here's your graph:

==============
x = 7, y = 5 --> (7,5)
--> is one circle
========================================
x = 1, y = -3 --> (1,-3)
is the 2nd circle
RELATED QUESTIONS
find an equation of a circle that tangent is to 3x+4y-15=0 at (5,0); contains (-2,-1) (answered by KMST)

Find an equation(s) of the circle(s) of radius 4 with center on the line 4x + 3y + 7 = 0... (answered by Alan3354,Fombitz)

tangent to 3x+4y-15=0 at (5,0) contains (0,4) find the equation of a... (answered by Alan3354,keyaytee)

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

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

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

Find the equation of a circle whose center is on 2x+y=1, tangent to 3x-4y=10 and radius... (answered by MathLover1,greenestamps)

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

Find the equation of the circle tangent to the line 4x � 3y + 12=0 at (-3, 0) and also... (answered by AnlytcPhil)