SOLUTION: Please help me with this! Show that the two circles (x - 3)^2 + (y-4)^2 = 25 and (x-1)^2 + (y-5/2)^2 =225/4. touch each other.

Algebra ->  Length-and-distance -> SOLUTION: Please help me with this! Show that the two circles (x - 3)^2 + (y-4)^2 = 25 and (x-1)^2 + (y-5/2)^2 =225/4. touch each other.      Log On


   

Question 992326: Please help me with this! Show that the two circles (x - 3)^2 + (y-4)^2 = 25 and (x-1)^2 + (y-5/2)^2 =225/4. touch each other.
Found 3 solutions by Edwin McCravy, ikleyn, Theo:
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
%28x-3%29%5E2+%2B+%28y-4%29%5E2+=+25
%28x-1%29%5E2+%2B+%28y-5%2F2%29%5E2+=+225%2F4
Subtract the two equations to get the equation of the line through
their point(s) of intersection if they have any points in common. 

%28x+-+3%29%5E2+-+%28x-1%29%5E2++%2B+%28y-4%29%5E2+-+%28y-5%2F2%29%5E2+=25-225%2F4

Factor first two and last two terms on the left as the
difference of squares:



%28x-3-x%2B1%29%28x-3%2Bx-1%29%2B%28y-4-y%2B5%2F2%29%28y-4%2By-5%2F2%29=-125%2F4

%28-2%29%282x-4%29%2B%28-4%2B5%2F2%29%282y-4-5%2F2%29=-125%2F4

%28-2%29%282x-4%29%2B%28-8%2F2%2B5%2F2%29%282y-8%2F2-5%2F2%29=-125%2F4

-4x%2B8%2B%28-3%2F2%29%282y-13%2F2%29=-125%2F4

-4x%2B8-3y%2B39%2F4=-125%2F4

Multiply through by 4

-16x%2B32-12y%2B39=-125

-16x-12y%2B71=-125

-16x-12y%2B71=-125

-16x-12y=-196

Divide through by -4

4x%2B3y=49

3y=-4x%2B49

y=expr%28-4%2F3%29x%2B49%2F3

This is the equation of the line through their
point(s) of intersection if the intersect. 

Substitute in

%28x-3%29%5E2+%2B+%28y-4%29%5E2+=+25

x%5E2-6x%2B9%2B+%28%28expr%28-4%2F3%29x%2B49%2F3%29%5E%22%22-4%29%5E2+=+25

x%5E2-6x%2B9%2B+%28expr%28-4%2F3%29x%2B49%2F3-4%29%5E2+=+25

x%5E2-6x%2B9%2B+%28expr%28-4%2F3%29x%2B49%2F3-12%2F3%29%5E2+=+25

x%5E2-6x%2B9%2B+%28expr%28-4%2F3%29x%2B37%2F3%29%5E2+=+25

x%5E2-6x%2B9%2B+%28expr%281%2F3%29%28-4x%2B37%29%5E%22%22%29%5E2+=+25

x%5E2-6x%2B9%2B+expr%281%2F9%29%28-4x%2B37%29%5E2+=+25

x%5E2-6x%2B9%2B+expr%281%2F9%29%2816x%5E2-296x%2B1369%29+=+25

Multiply through by 9

9x%5E2-54x%2B81%2B%2816x%5E2-296x%2B1369%29+=+225

9x%5E2-54x%2B81%2B16x%5E2-296x%2B1369+=+225

25x%5E2-350x%2B1450+=+225

Divide through by 25

x%5E2-14x%2B58+=+9

x%5E2-14x%2B49+=+0

%28x-7%29%5E2=0

That give the double solution x=7.  The double solution
means that the circles intersect at one point, where x=7.

Substituting in

4x%2B3y=49
4%287%29%2B3y=49
28%2B3y=49
3y=21
y=7

Therefore they intersect at the one point (7,7)



Edwin

Answer by ikleyn(52776) About Me  (Show Source):
You can put this solution on YOUR website!
.
Please help me with this! Show that the two circles (x - 3)^2 + (y-4)^2 = 25 and (x-1)^2 + (y-5/2)^2 =225/4. touch each other.
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The first equation is for the circle of the radius  r = 5  with the center at the point  (x,y) = (3,4).

The second equation is for the circle of the radius  R = 15%2F2 = 7.5  with the center at the point  (x,y) = (1, 5%2F2) = (1, 2.5).

The distance between the centers is  d = sqrt%28%283-1%29%5E2+%2B+%284-2.5%29%5E2%29 = sqrt%282%5E2%2B1.5%5E2%29 = sqrt%284+%2B+2.25%29 = sqrt%286.25%29 = 2.5.

Notice that the second circle is larger than the first one,  and the center of the first circle is located
inside of the second circle.  See the Figure below.

Now notice that R = d + r:   7.5 = 2.5 + 5.

It means that the first circle touches the second one from the inside.





Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
the two circles are:
(x-3)^2 + (y-4)^2 = 25
(x-1)^2 + (y-5/2)^2 = 225/4

if they touch, then the tangent line to each circle will pass through the point that they touch at.

the radius of each circle will be perpendicular to the tangent line at that point.

this means the two radii will form a straight line that goes through the point of tangency and will intersect both circles at the point of tangency.

the line will be a straight line formed by the line segment between the two centers of the cirfcle.

the centers of each circle are (3,4) and (1,(5/2)

using these two points, you get a straight line with the equation of y = 3/4 * x + 7/4.

that line should intereset with both circles at the same point.

i solved it graphically to see that the point of intersection is (7,7).

that graph is shown below:

$$$

it should be able to be solved algebraically but i didn't do it because i ran out of time and it's messy.

i did verify algebraically that the point (7,7) satisfies both equations, so there's no question that is the point of intersection of the two circles.


you would probably want to solve the intersection of that line with each equation using the substitution method.

graphing was much easier.