Finding the equations of circles

[SimonSpacz]

Hey guys, I'm having a bit of trouble trying to solve this one, could anyone offer some advice?
The question asks you to find two circles which satisfy the following conditions:
Radius of length 5, has the point (7,8) on it, and is at a tangent to the line y-2x+4=0

I played around with it for awhile, letting the radius equal to distance from centre to point, and perpendicular to the tangent but ended up getting nowhere.
Any help would be appreciated.

 

[Deleted member 4993]

Hey guys, I'm having a bit of trouble trying to solve this one, could anyone offer some advice?
The question asks you to find two circles which satisfy the following conditions:
Radius of length 5, has the point (7,8) on it, and is at a tangent to the line y-2x+4=0

I played around with it for awhile, letting the radius equal to distance from centre to point, and perpendicular to the tangent but ended up getting nowhere.
Any help would be appreciated.

You said you worked with the problem - show us those works (even if you think those are wrong).

Try to draw some possible situations. I do not see any unique solution (following the given problem statement)


You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/th...217#post322217

We can help - we only help after you have shown your work - or ask a specific question (not a statement like "Don't know any of these")

Please share your work with us indicating exactly where you are stuck - so that we may know where to begin to help you.

 

[SimonSpacz]

My work thus far

Okay I started off looking at the formula for a circle x^2+y^2+2gx +2fy +c = 0 for a circle of centre (-g,-f)
I plugged in the (7,8) and got my first equation 14g+16f+c=-113
Next thing was I took the distance from the centre to the point using the radius as the length and got:
g^2+f^2+14g+16f+88=0

Next I used the formula for perpendicular distance from a point to a line, using the tangent and centre, this gave me:

5√5 = |-2g+f-4| (modulus symbol, looks a bit unclear)

Now I'm wondering do I take both modulus answers, use these values for f/g and fill back into the first equation? I would end up with 4 different circles then I think? and the figures don't look too pleasant so that kind of shook me a bit. Can anyone elaborate/ help me reach a conclusion on this.
Thanks kindly

 

[DrPhil]

The question asks you to find two circles which satisfy the following conditions:
Radius of length 5, has the point (7,8) on it, and is at a tangent to the line y-2x+4=0

Okay I started off looking at the formula for a circle x^2+y^2+2gx +2fy +c = 0 for a circle of centre (-g,-f)
I plugged in the (7,8) and got my first equation 14g+16f+c=-113
Next thing was I took the distance from the centre to the point using the radius as the length and got:
g^2+f^2+14g+16f+88=0

Next I used the formula for perpendicular distance from a point to a line, using the tangent and centre, this gave me:

5√5 = |-2g+f-4| (modulus symbol, looks a bit unclear)

Now I'm wondering do I take both modulus answers, use these values for f/g and fill back into the first equation? I would end up with 4 different circles then I think? and the figures don't look too pleasant so that kind of shook me a bit. Can anyone elaborate/ help me reach a conclusion on this.

Musings.

I started by sketching the tangent line, and the point (7,8). The center (pardon my spelling!) of the circle has to be 5 units from the point, and also 5 units from the line. So I then drew a circle of radius 5 centered at (7,8), and drew a line parallel to and 5 units from the tangent line. The circle and the line intersect at two points, which are the centers of the two required circles. It is clear from the sketch that both of those points are in the first quadrant.

I don't see any problems with your approach. Using both signs for the modulus should give you the two centers.