SOLUTION: Find the equation of a circle whose center is on 2x+y=1, tangent to 3x-4y=10 and radius 5. thank you

Question 1130502: Find the equation of a circle whose center is on 2x+y=1, tangent to 3x-4y=10 and radius 5. thank you
Found 2 solutions by MathLover1, greenestamps:
Answer by MathLover1(20850) (Show Source): You can put this solution on YOUR website!

find intersection of line and tangent
.......solve for

---------------------

substitute in

then,

intersection point is:
P( ,)

-> since center is on a line , we have

=>
so, center C (,)=(,)
distance from and is

or

=>

or

so, we have two centers that lie on given line:
center C (, )
and
center C (,)
then, we have two circles:

or

Answer by greenestamps(13203) (Show Source): You can put this solution on YOUR website!

The solution by tutor @MathLover1 is not correct. Her solution used the intersection of the lines 2x+y=1 and 3x-4y=10 as the point of tangency, which is not right. Her graph shows that her two circles are not quite tangent to the line 3x-4y=10.

Note that the intersection of the two lines would be the point of tangency of the two circles only if the two lines were perpendicular to each other; but they are not.

We need to find point(s) on the line 2x+y=1 that are 5 units from the line 3x-4y=10.

Solve the equation of the line 2x+y=1 for y to get y = -2x+1; then an arbitrary point on that line has coordinates (a,-2a+1).

The distance of a point (p,q) from the line with equation ax+by+c=0 is

We want the distance from the point (a,-2a+1) to the line 3x-4y-10=0 to be 5:

OR

OR

OR

OR

We have two circles, one on each side of the line 3x-4y=10. The x coordinates of the centers of the circles are a=-1 and a=39/11; the corresponding y coordinates are -2a+1 = 3 and -2a+1 = -78/11+1 = -67/11.

So there are two circles with radius 5 and with centers on the line 2x+y=1 that are tangent to 3x-4y=10.

(1) center (-1,3):
(2) center (39/11,-67/11):

Here is a graph:

You can see that the two points of tangency are on either side of the line 2x+y=1.
RELATED QUESTIONS
Determine the equation of the circle whose radius is 5, center on the line x=2 and the... (answered by Fombitz)
6. A circle of radius 5 has its center on the line 3x � 2y + 6 = 0 and the circle is... (answered by jim_thompson5910)
Please help me with this question. Find the equation of a circle, whose center is (3,1)... (answered by Alan3354)
. Find the center and radius of the circle whose equation is x2 + 2x + y2 � 4y + 1 = 0. (answered by stanbon)
Find an equation(s) of the circle(s) of radius 4 with center on the line 4x + 3y + 7 = 0... (answered by Alan3354,Fombitz)
Find an equation(s) of the circle(s) of radius 4 with center on the line 4x + 3y + 7 = 0... (answered by Fombitz)
how do you find the center-radius form of the equation of a circle with center (4,-7) and (answered by solver91311)
The circle is tangent to y-axis, and the center is at (5, 3). Find the equation of a... (answered by dkppathak)
A circle is tangent to the line 3x-4y -4 = 0 at the point (-4,-4) and the center is on... (answered by robertb,greenestamps,ikleyn)